Ex 2: Simplify a Trigonometric Expression Using Negative Angle Identities

Ex 2: Simplify a Trigonometric Expression Using Negative Angle Identities


– WE WANT TO SIMPLIFY
THE GIVEN TRIG EXPRESSION TO A SINGLE TRIG EXPRESSION
WITH NO FRACTIONS. SO TO BEGIN, WE’LL APPLY
THE NEGATIVE ANGLE IDENTITIES GIVEN HERE BELOW. SO COSINE -X=COSINE X. SO WE HAVE COSINE X AND THEN
+ SINE -X=NEGATIVE SINE X. SO WE HAVE NEGATIVE SINE X
DIVIDED BY COTANGENT -X=NEGATIVE COTANGENT X. NOTICE HERE WE HAVE A NEGATIVE
DIVIDED BY A NEGATIVE. SO THAT WOULD BE POSITIVE. LET’S ALSO WRITE COTANGENT
IN TERMS OF SINE AND COSINE. SO COTANGENT X WOULD BE EQUAL
TO COSINE X DIVIDED BY SINE X. SO LET’S WRITE THIS AS COSINE
X, AND THEN WE’LL HAVE +. THE NUMERATOR IS STILL SINE X. SO WE HAVE SINE X, AND THEN WE HAVE DIVIDED
BY COSINE X DIVIDED BY SINE X. HERE WE HAVE
A COMPLEX FRACTION, BUT REMEMBER, A FRACTION BAR
REPRESENTS DIVISION. SO NOW, WE CAN WRITE THIS
AS COSINE X +. LET’S WRITE SINE X
AS SINE X/1, AND THEN WE HAVE DIVIDED
BY COSINE X DIVIDED BY SINE X, AND INSTEAD OF DIVIDING, LET’S
MULTIPLY BY THE RECIPROCAL. SO NOW, WE’LL HAVE COSINE X
+ SINE X/1 x SINE X/COSINE X. NOW, WE’LL GO AHEAD
AND MULTIPLY. LET’S GO AHEAD AND WRITE
COSINE X AS COSINE X/1. SO WE HAVE COSINE X/1
AND THEN +. THIS WOULD BE SINE SQUARED
X/COSINE X, AND NOW, TO ADD
THESE FRACTIONS, WE NEED TO OBTAIN
A COMMON DENOMINATOR, WHICH WE CAN SEE
WOULD BE COSINE X. SO WE’LL MULTIPLY COSINE X/1
BY COSINE X/COSINE X. SO NOW, WE HAVE A COMMON
DENOMINATOR OF COSINE X. THE NUMERATOR IS NOW COSINE
SQUARED X + SINE SQUARED X, AND FROM HERE
WE SHOULD RECOGNIZE OUR PYTHAGOREAN IDENTITY, COSINE SQUARED X
+ SINE SQUARED X=1. SO THIS SIMPLIFIES TO 1
DIVIDED BY COSINE X, BUT OUR DIRECTIONS SAY
WE DON’T WANT FRACTIONS, AND SINCE SECANT AND COSINE
ARE RECIPROCALS OF ONE ANOTHER, 1 DIVIDED BY COSINE X
=SECANT X. SO THE GIVEN EXPRESSION
SIMPLIFIES NICELY TO SECANT X. I HOPE YOU FOUND THIS HELPFUL.  

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