SOLUTION: This is a Trigonometric Identity Problem. Simplify the expression: csc(-x)-csc(-x)cos^2(x) I think that the two csc(-x)s cancel out to leave the answer being cos^2(x) but I'

Algebra ->  Trigonometry-basics -> SOLUTION: This is a Trigonometric Identity Problem. Simplify the expression: csc(-x)-csc(-x)cos^2(x) I think that the two csc(-x)s cancel out to leave the answer being cos^2(x) but I'      Log On


   

Question 307231: This is a Trigonometric Identity Problem.
Simplify the expression: csc(-x)-csc(-x)cos^2(x)
I think that the two csc(-x)s cancel out to leave the answer being cos^2(x) but I'm not sure since they are multiplied.
Please confirm my answer.
Thanks!
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Rules for trig functions of negative values:

The cosine and secant are the only even trig functions,
and the negative sign "disappears":

cos%28-theta%29+=+cos%28theta%29
sec%28-theta%29+=+sec%28theta%29

All the other trig functions are odd trig functions,
and the negative sign comes out in front:

sin%28-theta%29+=+-sin%28theta%29
tan%28-theta%29+=+-tan%28theta%29
csc%28-theta%29+=+-csc%28theta%29
cot%28-theta%29+=+-cot%28theta%29

csc(-x)-csc(-x)cos2(x) becomes

-csc(x)-[-csc(x)]cos2(x) 

-csc(x)+csc(x)cos2(x)

Factor out -csc(x)

-csc(x)[1-cos2(x)]

Us the identity Sin%5E2theta%2BCos%5E2theta=1 written as Sin%5E2theta=1-Cos%5E2theta to replace the bracketed expression:

-csc(x)sin2(x)

Use the identity csc%28theta%29=1%2Fsin%28theta%29 to replace the cosecant factor:

-1%2Fsin%28x%29sin2(x)

This simplifies to

-sin(x)

Edwin