# Trigonometry Proofs Involving Half and Double Angles

BAM! Mr. Tarrou. Welcome to my third video

about dealing with half angle and double angle identities. My goal in this video is to start

working on some and show how half angle and double angle identities they work inside some

proofs or verify some identities. I am going to do four examples and hopefully get done

in the time span of this video. So let’s get started. The sine of two theta is equal to

two cotangent of theta over one plus cotangent squared of theta. Ok. Well, the sine of two

theta you might know, you should know if you have your memorized and if not start working

on it or at least look at your formula sheet, is two times sine theta cosine theta. So I

need to have all of this written in terms of sine and cosine and do some algebra and

see what happens to get the two sides to match. Well, what to do…what to do. The first thing

to do is go BAM! this is cosine over sine and this is cosine over sine, find common

denominators and do all that good stuff, and it is going to look something like this. Two

cosine theta over sine theta over one plus cosine squared theta over sine squared theta.

We need a common denominator so we are going to take one and make it sine squared theta

over sine squared theta. That is going to become two cosine theta over sine theta all

over sine squared theta plus cosine squared theta over sine squared theta. Do you know

your identities? Sine squared plus cosine square is equal to one and any time you have

one fraction over one fraction you can flip that bottom up. We are going to flip the bottom

up and write two cosine theta over sine theta times sine squared theta when we flipped the

bottom up, and don’t forget that sine squared plus cosine squared equals one. The sines

will cancel out. This one sine in the denominator will cancel out with one of the two sines

in the numerator. Remember you cancel out factors but not terms. We get two times sine

theta cosine theta. I am just turning the multiplication around to perfectly match the

identity. That is ok because three times five and five times three is fifteen. This is the

sine of two theta. Now, if you are really at proofs and you just stumbled upon this

video or you are just getting used to them, or you are using this to teach yourself how

to do these. There is a reason the back of your book says that proofs may vary. Because,

well they may vary:) That is why like teaching trig proofs so much, because there is no just

use these five steps and if you follow them in a certain order you get the right answer.

You have to think a bit more. Play with the algebra, do some substitution, see what helps,

and for goodness sakes if you have a pythagorean identity that is going to help you right off

the bat…maybe you should use it. So, I have shown you a proof where I immediately went

into sine and cosine and I did an ok amount of work. That is fine and I get the right

answer. You know what, this is full credit…BAM! Well, here is another way. Somebody might

be going, why didn’t you do it like this? And you are going to go, because did it MY

WAY!!! One plus… I know I am bad at singing:) One plus cotangent squared, let’s write this,

two cotangent theta over one plus cotangent squared is cosecant squared. Now when I turn

everything in into terms of sine and cosine, I will not have to find common denominators

and I will be done much quicker. This is still going to be two times cosine over sine, so

it is two cosine theta over sine theta. Cosecant squared is one over, it is one of the reciprocal

identities, that is one over sine squared theta. Look, see I originally had sine squared

plus cosine squared there after finding common denominators and after you applied that first

Pythagorean Identity, so I kind of needed to know those Pythagorean Identities any way

regardless of how I do this proof. Any way, I am rambling. The denominator is going to

get flipped up becoming sine squared theta over one. Once again, the sines will cancel

out and once again we have the sine of two theta. So, I did one proof two ways. Is that

two examples or is that one? I don’t know:) But, when you are doing proofs, when you are

verifying identities you want to… Well, look at the two sides and see what is going

on, are the angle measures the same, are the degrees the same. The only way you can change

the degree is through factoring and canceling (or Power Reducing Identities). The way you

can change the size of the angle measures is the substitution of the half angle or double

angle identities (or sum and difference identities) or at least that is all I am covering. See

if there is any Pythagorean Identities or any of those heavy duty identities whether

it is sum or difference, half angle or double angle, power reducing, whatever that is going

to help you right off the bat, then go to turning everything into sine and cosine…not

as a last resort because that is a really common skill or process. But sometimes you

can shorten your proof up by applying some identities early on if there is any there

to help you. Ok, I have a feeling this is going to be a two part lesson for proofs.

How about we take a look at the cosine squared of theta over two, and let’s see if we can

prove that this equals the secant of theta plus one all over two secant theta. Well,

cosine of theta over two is the square root of one plus cosine theta over two, not anything

to do with secant. So there are no identities that are going to help me immediately, no

heavy duty identities, there is no algebra to do, so we are going to have to go into

sine and cosine and hope that the algebra allows us to get the two sides closer and

help this click so that we can see what is going to happen. So secant is equal to one

over cosine theta. One, I am going to make common denominators now. One is going to be

rewritten as cosine theta over cosine theta, common denominator, over two times one over

cosine theta or just two over cosine theta. I am going to put the top together and get

one plus cosine theta all over cosine theta over two over cosine theta. Any time again

you have one fraction over one fraction, you want to flip that denominator fraction up

next to the numerator with multiplication. Any time your denominators are an exact perfect

match, they are going to cancel out. So, we have…wrong color…one plus the cosine of

theta over two. Hmm. Ok, so you could stop there or you can keep manipulating. If you

are not sure how to make this equal to that and you are totally stuck, then maybe just

playing with the other side. You know, what is the cosine of theta over two? Well, it

is the square root of one plus cosine theta over two, and then there is a power of two

up here. See it? So that means that it is the square root of one plus cosine theta over

two in that square root, squared. Well, that power of two is going to cancel out the radical

and if your teacher wants to only see the work on one side of the proof, then you can

sort of manipulate both sides on your scratch work, then anything on this other left hand

side you can go in reverse on the right hand side until you get cosine squared theta over

two. As far as I am concerned, the left and right hand sides match and I am done. BAM!!!

ALL RIGHTY THEN, let’s do another one:) How about one that is pretty straight forward

and then we will do a hard one. Tangent of x over two equals, you might think that these

are all hard, cosecant of x…if this is your first time learning them…minus the cotangent

of x. Ok, if you look at your formula sheet, you will notice that the tangent of a half

angle identity only has sine and cosine in the three versions of that identity. So, no

cosecants and cotangent and we are going to work on the right hand side. Remember, and

also I have not said this in this video, when you are work on proofs you always want to

simply. Work down the tree, work your way down to the main trunk. Don’t things more

complicated and have a thousand branches you can go up to. Ok, whatever, we are going to

make this in terms of sine and cosine. Cosecant is one over sine x. Cotangent is cosine x

over sine x. Well, shoot…we are done. One minus cosine x all over sine x is one of the

half angle identities for tangent. Also don’t stop until the two sides are an exact match

please! Do work your proofs straight down using nice and clear hand writing, no scribble

all over the place. Your teacher, your test papers are giving you the start and finish

of the problem, we have to grade everything in the middle. We have to see every single

step, make sure that your algebra is right and you logic is correct. We are not just

looking at the answer. We gave you the answer. Make sure that your test is gradable, otherwise

you may not like what you get back. Ok, at least my kids won’t. I am not going to try

to decipher their hieroglyphics. The sine of three x over the sine of x

minus the cosine of 3x over the cosine x and

all of this is supposed to equal two. I hope I don’t have to erase anything on my board

because this is going to be a lot of writing though not a very long proof. Well I guarantee

you that some of my students are going to want to say that 3x over x makes 2x, or the

x’s cancel out, or the sines cancel out. This is not sine times three times x, it is not

cosine times three times x, the cosine and sine are math functions. It is the sine of

(3x) and the cosine of (3x), and no nothing is going to cancel. Alright, so get that out

of your head. We want to prove that this equals two. Man, when I look at my formula sheets

I don’t have three x’s. I have half angle and double angle identities. So, this must

also include some sum and difference identities. I am going to rewrite this so that it is the

sine of x plus two x, that is equal to the sine of three x, minus cosine of x plus two

x, and our denominators of sine x and cosine x. I am not even going to mess with the right

hand side. I think I am going to need a lot more room, so I am going to work down and

possibly need some room over here. I am going to apply the sum identity for sine. It is

the sine of the first times the cosine of the second plus cosine of the first times

the sine of the second minus, don’t forget the parenthesis because I am going to write

all this…I am not going to do that right now…all of that is over the sine of x. This

is minus, applying this sum identity, cosine x times cosine of 2x, don’t forget with your

cosine function with sum and difference identities to change that middle sign, so minus sine

of x times sine of 2x all over cosine of x. Now I have got two humungous fractions that

need to come together and clearly a lot of stuff is going to cancel because the answer

is two, right. We need a common denominator so I am going to multiply this entire fraction

by cosine on the top and bottom

and sine x on the top and bottom. WHOOO!!! Man, now I have to write an even longer sentence.

Ok, so we have…let me get down here… we have cosine x times sine x times cosine 2x

plus cosine times cosine is cosine squared x times sine 2x. Now I am making common denominators

here, so I am going to put a minus sign and a parenthesis because there are two terms

in the second numerators. Sine x times cosine x times cosine 2x. I am getting really quiet

here because I don’t want to make a sign error here. All of that is over cosine x sine x.

Ok, now what do we have here. We have cosine x sine x cosine 2x, cosine x sine x cosine

2x, this one is positive and after we distribute the negative sign that is going to be positive.

So, cosine x sin x cosine 2x is going to cancel with minus sine x cosine x cosine 2x. That

is going to save some writing! We got cosine squared x times sine of 2x. I am going to

need to go up here now. I hate jumping around like this. I hope that you can follow it.

Cosine squared x times

sine 2x, negative times negative is positive sine squared x times sine of 2x all over the

denominator cosine x times sine x. Good thing… I might still need to erase part of this.

You might want to copy what I am doing here. Got it? Ok, so I probably should erase the

holiday lights for this. We have got the sine of 2x and the sine of 2x in both terms and

we are going to factor that out. We are going to pull this out and get the sine of 2x times

cosine squared x plus sine squared x. This is all over cosine x times sine x. Are you

starting to feel it. I am getting excited! We are almost done. Cosine squared x plus

sine squared x, that is the first Pythagorean Identity and that is equal to one. Now we

have the sine of 2x over cosine x times sine x. That becomes…. We have a double angle

and that is a single angle, so nothing is going to happen until we fix that. This is

sine of 2x is two sine x cosine x all over cosine x sine x. The cosines cancel out, the

sines cancel out, and I just did all that work to get an answer of two. Are you kidding

me?! BAM!!! Mr. Tarrou. Go Do Your Homework!

I am the one who is glad I found your channel, of course I will share your vids to any friends. Thank you ^_^!

Great job on that 90 grade by the way…And THANK YOU for sharing:)

i'm searching topics about domain and range of sine and cosine…

I would start with my video about understanding the basic graphs of sine and cosine and/or look at graphing sine and cosine without a calculator. The domain of Sine and Cosine is all real number, so you only have to worry about range. If you know the amplitude and the vertical shift you will be able to figure out the range.

okay thanks.

BAM!!! When hes done you want need anymore chalk!!!!

Don't worry…I have chalk for years to come!!!

Thanks for liking and subscribing too:)

I hope you went into class and passed that exam like BAM!!!

This is great stuff! We need more teachers like you. Keep the lessons coming, you've been saving my butt in trig class.

Thanks for the compliment and THANK YOU for choosing my channel to learn from! I hope you are sharing it with all your friends and classmates…remind them that it is important to like and subscribe to help educational channels like mine to groW and remain FREE!

Thank you sooo much for this, I really wish you were my math teacher… My teacher is boring as hell… 🙁

Hahhaa,,, thank you! I've got a test today and I this helped a bit!

Test tomorrow. ..this helped…I think I would've cried trying to do that problem! Thankssss

You're such a bro. This helped so much

I really love when he brakes his chalks like that at the ending of his vids haha

We need IMO

The last question could be done much easier boss by taking sinxcosx. as denominator, the numerator will be sin3xcosx-cos3xsinx= sin (3x-x)= sin 2x. The bottom sinxcosx=1/2 sin2x,

Sin2x/ 1/2sin2x= 2 QED

i keep replaying the smashing of the chalk at the end lolol xD thanks!

You have helped me remember everything I forgot over the semester and now I'm not so worried about my final exam

your videos are awesome… I'm taking an online trig class and strictly watch your videos to study. Throw that chalk and walk off like a boss!

AMAZING VIDEO!!!!!!! (sorry for the caps) LOL

Did all that work for an answer of 2?!?!?!

destroys chalk on the boardI just about died laughing, but your videos are helpful, but I feel like the practice problems might be too easy.Great videos….they always help me. Thanks

About to rock out on today's trig exam.. like BAM.

Thanks as always Prof. Rob.

Thanks tons for all of your help! Feels like I have my own personal tutor at the click of a button! Really helps me grasp the information.

Appreciate all the help Prof. Rob. You da best!

Another chalk throw and a "do your homework" jingle! These videos keep getting better and better. Just tested today and felt extremely confident with the outcome. Really enjoyed the meaty proof you closed out this tut with, I think my teacher goes a little easy on us sometimes. Wish I had a instructor like you in High School. Thanks again PRB!

I wish you could be my teacher! I learned more in this 17 minute video than i did 2 weeks in my Trig class.

You should teach professors how to teach/instruct/lecture. Your videos are never boring and never lack enthusiasm or value. Could you do a video on this stuff using sec,csc, and cot

BAM just aced trig final because of these awesome videos, now on to calc

After converting the top fraction and bottom fraction into cos's over sine's I multiplied the top fraction and the bottom by sin^2x and did the cancelling until I got 2cosxsinx/1. Did I follow the laws of algebra correctly??

Did anyone else catch the Ace Ventura reference? haha

I hear you say, "Go do your home work, " after several of your videos. Is there any practice questions somewhere?

Wish you could be my teacher!

Our teacher is too fast and doesnt set specifics and is quite rather trivial.

@ProfRobBob Didn't you get rid of the negative sign in between both fractions? Why so you distribute the negative sign when you move the equation up?

nehahahaha u r amazing

You are AWESSSSSSSSSSSSSSSOMEEEEEEEEEEEE. Please come teach at Citrus College here in California. LOL . 😀

Great video! Keep it up,I need you jeje 👌👍👏

OH MY GOSH THIS VIDEO GOT MY HEART RATE GOING ALL FUNKY WOW PROOFS ARE AMAZING I LOVE LIFE

I bet you go through a lot of chalk lol

You are an actual life saver :)) i was worried sick that i would do horrible on my trig test but after watching your videos for 2 hours i feel like im finally prepared. Thank you so much!

You make me cry thank you so much!!

hahahahaha

how to prove: tan3xtan2xtanx=tan3x-tan2x-tanx

really helped

wooow woow woow.. IF U'RE MY PROF, I WILL NEVER FEEL SLEEPY THEN! YOU'RE THE BEST! #BAM!! xD

Wow! Thank you for showing us every single step! I'm about to ace this test that I'm taking in few days! #BAM hahaha

I like your videos but that's not the right use of the word bam or bayyyym if your American. It's not necessary.

Awesome Professor Rob, as always

Holy crap! Best teacher out there. Im in college taking trig as a pre req and my professor is the worst. I've used these videos to teach myself and have made A's in all my tests:) thank you!

17:27!!!! I wish we had more teachers like this!! You really do put the joy in learning math!!! Thank you again for all your work!

For the first example, I saw 1 +cot squared x as cscx. Thanks for this video. I just love trig identities. It was great learning half angles from you as I barely understood it when I first learned it.

Thank you so much for patiently explaining everything, Sir. 🙂 There are no students who are terrible in math but are simply oblivious of what they are supposed to do since the professor won't directly tell the guidelines in solving such. I'm surviving my semester with your videos!

You've just made my day! I was struggling with the cos^2 (x/2)

I tried to figure it all out by myself but ended up crying (not really) but BAM! I saw this vid and thanked the math gods for you sir! cheers!

"I'm not gonna try to decipher their hieroglyphics." Hahaha you're the best xD

"cosines cancel out, sines cancel out.. and I just did all that work to get an answer of TWO!? Are you kidding me!?

smashes chalkWOOO BAM!!" hahaha, i laughed so hard.Half angle and double angle Trig proofs is now Closed Captioned! #math

I DO NOT MISS high school at all. I can't believe I used to know this enough to graduate with honours. Well kids I went on to University and had a successful career and never used this stuff again. You however are learning it NOW. If you want a career in computers or the sciences YOU WILL NEED THIS. Work hard and remember it.

i got a different approach on this proof also but BAAAAM! go to your homework haha..

Hi Prob Rob Bob, I would like to ask this cosx(sinxcos2x+cosxsin2x), why did you not multiply cosx with cos 2x? and also cosx with sin2x?

Your videos helped me a lot. I've been watching almost all your trig videos and BAM! You're great! Thank you.

i also smash my pen just as what he did when i solved one of our identites but i got kick out of my room :'(

how did the 2cossin = 2sin?

Dude your a lifesaver. Very easy to understand as well as the fact you are entertaining to watch and as a result, i stay engaged in what you are teaching.

Excellent video.

Smashing the chalk was your best exit yet.

drop the mic, put you cape on, and walk off the stage.

How does tanx/2 = 1-cosm/sinx? What identity is it?

Hello there! I wanted to let you know that I watched all of your videos on half and double angles and sum and difference formuals to prepare for a quiz in precalculus. Thanks to your videos, I thought the quiz was very easy, despite the number of students in my class who thought it was our hardest quiz all year. I will let them in on my secret haha. Thank you for your videos 🙂 I am so happy I was able to find the best math content on YouTube.

How many sticks of chalk do you need to buy to replace the shattered ones? XD jk… But you videos really do help… I have a test tomorrow and i've been watching your videos for the past few days, and I'm feeling confident. Thank you for helping me learn trig! I couldn't have figured it out without you 🙂 Keep up the good work because it really is helpful

thanks you for this video 🙂

5 years and it still helps me and I've never seen a teacher so excited about math lol

At 12:17, why did you split the 3x into 2x+x and when do we do this?

cool

you make math a lot of fun , you changed my perspective of mathematics since I started watching your videos ,thank you very much

I wish I had a math teacher as enthusiastic as you are in school. Thanks for making videos! I really like how you take the time to explain what you are doing and why. Subbed!

You are a math God. Thank you for these! They are saving me currently. Nothing this semester has made as much sense as your videos do. I think I might just watch these in class instead??

I love your lecture. I whish you were my teacher

"I just did all that work just to get an answer of 2, are you kidding me?"

-Every highschool student ever =)

BAM!!!!! i wish my 75 year old teacher smashed chalk into smitherins against the board with a BAMM!!! She's a good teacher… but she's up there in age if you know what i mean.

That ending was boomshot, you have been very helpful keep it up

Thank you so much! I lost my notes and my exam is tomorrow you saved my grade!

My teachers would be jealous that you have enough chalk that you can just throw at the wall 😂😂😂. Anyways amazing job on the videos! They are a live saver for my Pre-Calc class. I don't know what I would do without them!

Epic ending! haha

Love your activeness

Hahahah you're so funny

Is the proof of sin2theta=2sinthetacostheta just sin (theta + theta) or is there more to it?

I would like to take a moment to thank you for your fantastic work sir !

I depend on your explanations very much and your precious help is very highly appreciated by me !!

Absolutely phenomenal !

Keep it up !

Thought I'd make a proof of my own:

tan (a ± b) = (tana ± tanb) / (1 ∓ tanatanb); tanx = sinx/cosx

This one was tricky at first.

Prove: (tana ± tanb) / (1 ∓ tanatanb) = (sinacosb ± cosasinb)/(cosacosb ∓ sinasinb)

Multiply by secasecb/secasecb

= ((sinacosb ± cosasinb)(secasecb))/((cosacosb ∓ sinasinb)(secasecb))

Distribute

= (sinacosbsecasecb ± cosasinbsecasecb)/(cosacosbsecasecb ∓ sinasinbsecasecb)

cosxsecx = 1, where cosx =/= 0; which shouldn't be a problem in this case given quotient rule

= (sinaseca ± sinbsecb)/(1 ∓ sinasecasinbsecb)

sinxsecx = tanx (which also works inverted: cosxcscx = cotx); What I call the trigonometric product rule

= (tana ± tanb)/(1 ∓ tanatanb)

You're Great SIR . You made me understand the graphs of trig functions . i was not able to imagine about the domain and ranges but you are the one who cleared my concepts . You take your precious time out to share your knowledge with us which deserve a applause. We all RESPECT you. I Subscribed your channel and wish to see more new videos from you Sir!

Sin^2(theta) +sin^2(theta) = sin^2(theta) ? This is at 1:52 and I am confused how you added sin^2(theta)/sin^2(theta) and got what you got.

As a High School Student sitting for Cambridge International Exams, Sir you deserve more than a like button after 3 long years I finally understand this part of trigonometric Identities and now i will follow this path in exams and continue to work hard….Your explanation are simple and very promising…Keep it up, you will make your way even higher than expected SIR…..

I have a problem with 7x can I split it to 3x+4x or do I have to split is up more?

That chalk bit is the equivalent of smashing a guitar after rocking out

Love the chalk explosion at the end 😂 now I won’t fail the test tomorrow!

Thank you for the amazing video!! I am so grateful that you took time to show me how to do analytic trig! I have a test this monday and this helps me feel prepared.

man watching this is like watching magic I swear

You make it look too easy.

Bam😂😂😂

Wow this is so Amazing,,,

Bro..for the last question i just found a simpler way.

Sin 3x= -Sin x and Cos 3x= -Cos x

So i subtitute it into the equation,

-Sin x/Sin x – (-Cos x/Cos x)

-1-(-1)

2