Expert: Socrates - 11/30/2009

Question
I have a question for Trig Identities.

I need to prove the Identity

(cosx / sec x -1) - (cosx / sec x + 1) = (2cosx/tan^2x)

Answer
cosx/(secx - 1) - cosx/(secx + 1)

multiply numerator and denominator of both fractions by cosx

= (cosx)^2/(1 - cosx) - (cosx)^2/(1 + cosx)

combine fractions over the common denominator (1-cosx)(1+cosx) = 1-(cosx)^2 = (sinx)^2

= ((cosx)^2 + (cosx)^3)/(sinx)^2 -  ((cosx)^2 - (cosx)^3)/ (sinx)^2

= 2(cosx)^3/(sinx)^2

= 2cosx (cosx/sinx)^2

= 2cosx (1/tanx)^2

= 2cosx/tan^2x