Question

Prove the identity
1+cscxsecx=cos(-x)-csc(-x)/cosx

Answer 1

cos(-x) = cos(x) because cos(x) is an even function meaning replacing x with -x gives the same answer.
csc(-x) = -csc(x) because csc(x) is an odd function meaning replacing x with -x gives the negative.
Therefore the right side can be written as:
{ cos(x) + csc(x) } / cos(x) = cos(x) / cos(x) + csc(x) / cos(x) = 1 + csc(x)sec(x) which is the left hand side. So we have proven the identity.

Answer 2

Thank you so much!