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X
- 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.