# Verifying Identities: Sum, Difference, Double, and Half Angle Identities

– WELCOME TO ANOTHER VIDEO ON VERIFYING TRIGONOMETRIC

IDENTITIES. THIS VIDEO WILL FOCUS

ON THE SUM, DIFFERENCE, DOUBLE

AND HALF ANGLE IDENTITIES. SO HERE’S A LIST OF ALL

THE IDENTITIES THAT WE’LL BE WORKING WITH

IN THIS VIDEO. ONE OF THE MOST CHALLENGING

PARTS ABOUT VERIFYING IDENTITIES USING THESE IS DECIDING

WHICH ONE TO USE AND SO SOMETIMES IT DOES TAKE

A LOT OF PRACTICE TO GET GOOD AT VERIFYING

IDENTITIES SO HOPEFULLY YOU’LL KEEP

A POSITIVE ATTITUDE AND STICK WITH IT. SO LET’S GO AHEAD

AND GET STARTED. WE HAVE A CHOICE HERE TO EITHER

USE THE DOUBLE ANGLE IDENTITY OR WE COULD MULTIPLY THIS OUT

ON THE LEFT. I’M GOING TO GO AHEAD

AND MULTIPLY THE LEFT SIDE OUT. SO WE’LL FOIL THIS. WHEN WE MULTIPLY THIS OUT

WE’LL HAVE SINE “A” x SINE “A” OR SINE SQUARED “A”. NEXT, WE’LL HAVE A SINE

“A” x COSINE “A”, AND THEN WE HAVE A COSINE “A”

x SINE “A.” SO WE HAVE TWO LIKE TERMS

SO WE’LL HAVE + 2 SINE “A” COSINE “A”

+ COSINE SQUARED “A.” THE NEXT THING WE SHOULD NOTICE IS SINE SQUARED “A” + COSINE

SQUARED “A” IS EQUAL TO 1. SO WE’LL HAVE 2 SINE “A” COSINE

“A” + 1 EQUALS SINE 2A + 1 AND NOW BELIEVE IT OR NOT

WE HAVE IT. REMEMBER THE IDENTITY SINE 2A IS

EQUAL TO 2 SINE “A” COSINE “A” SO WE CAN REPLACE THIS WITH SINE

OF 2A. SO IT IS IMPORTANT

THAT YOU ARE PRETTY COMFORTABLE WITH THESE IDENTITIES BECAUSE IT

WILL HELP YOU RECOGNIZE WHEN TO PERFORM THE CORRECT

SUBSTITUTIONS. LET’S GO AHEAD AND TRY ANOTHER. HERE ON THE RIGHT

THERE’S NOT MUCH WE CAN DO WITH COSINE X – SINE X BUT NOTICE ON THE LEFT

WE HAVE COSINE OF X + PI/4 WE CAN EXPAND THIS USING THE SUM

IDENTITY FOR COSINE. SO THE LEFT SIDE

CAN BE REWRITTEN AS THE COSINE OF “A” x COSINE B. SO COSINE OF X x COSINE OF PI/4. SINCE WE HAVE A SUM, WE’LL USE

A DIFFERENCE ON THE LEFT. SINE “A” OR SINE X x SINE B

OR SINE PI/4. NEXT, WE CAN EVALUATE COSINE

PI/4 AND SINE PI/4. IF YOU DON’T HAVE YOUR UNIT

CIRCLE HANDY THE SINE OF PI/4 AND THE COSINE

OF PI/4 ARE BOTH EQUAL TO THE SQUARE

ROOT OF 2/2 AND NOW WE’RE MAKING VERY GOOD

PROGRESS. YOU CAN SEE THAT THEY’RE ALMOST

THE SAME NOW. BOTH OF THESE TERMS

HAVE A COMMON FACTOR OF SQUARE ROOT 2/2 SO WE CAN FACTOR THAT OUT

AND I BELIEVE THEY’RE GOING TO MATCH NOW

AND THEY DO SO WE’RE DONE AND LET’S GO AHEAD

AND TAKE A LOOK AT ONE MORE. NOW THIS PROBLEM CAN LOOK

INTIMIDATING ESPECIALLY IF YOU’RE NOT FAMILIAR

WITH YOUR IDENTITIES. SO IT IS IMPORTANT THAT YOU HAVE

A GOOD LIST OF THEM AND YOU KNOW HOW TO USE THEM, AND ON THIS PROBLEM I’M GOING TO

WORK FROM BOTH SIDES. SO I CAN REPLACE SINE OF 2A

WITH 2 SINE “A” COSINE “A” AND ON THE RIGHT SIDE I’M GOING TO USE THE HALF ANGLE

IDENTITIES. NOTICE THAT THIS IS COSINE

SQUARED A/2 SO THAT’S GOING TO ELIMINATE

THE SQUARE ROOT HERE. SO COSINE SQUARED A/2 IS GOING

TO EQUAL 1 + COSINE A/2. AGAIN, SINCE THIS IS SQUARED IT’S UNDOING THAT SQUARE ROOT

– SINE SQUARED A/2 WE’LL HAVE 1 – COSINE A/2. NEXT, ON THE LEFT SIDE

WE HAVE A COMMON FACTOR OF 2 AS WELL AS SINE “A”

SO ON THE LEFT WE’RE LEFT WITH COSINE “A.” NOW ON THE RIGHT NOTICE

WE HAVE A COMMON DENOMINATOR SO LET’S GO AHEAD

AND ADD THESE FRACTIONS. NOW WE DO HAVE TO BE CAREFUL

WHEN WE DO THIS. WE’RE SUBTRACTING THIS ENTIRE

QUANTITY. SO LET’S PUT OUR PARENTHESES

IN PLACE. THE DENOMINATOR IS 2. NEXT WE HAVE 1 – 1 THAT’S 0 AND THEN WE HAVE COSINE

“A” – A -COSINE “A” SO THAT WILL GIVE US

2 COSINE “A” AND BELIEVE IT OR NOT

IN JUST A COUPLE OF QUICK STEPS WE HAVE VERIFIED THIS IDENTITY. THESE TWO SIMPLIFY NICELY AND SO WE HAVE COSINE

“A”=COSINE “A.” AGAIN, BECAUSE I PROVIDED

THE CORRECT IDENTITIES FOR EACH OF THESE PROBLEMS IT MAY SEEM A LOT EASIER

THEN WHEN YOU OPEN A TEXTBOOK AND START WORKING SOME PROBLEMS. SO YOU JUST HAVE TO BE PATIENT

WITH IT AND TRY DIFFERENT THINGS AND BECOME FAMILIAR WITH THOSE

IDENTITIES. AGAIN, THANK YOU FOR WATCHING

AND HAVE A GOOD DAY.

you actually helped me a lot!!! thanks !!

Thanks, this helped

You have as much energy and enthusiasm as Toby Flenderson