Trig Identities

Question

I am unsure of how to verify this using trig identities...

(1)/(1+sec(A))+(1)/(1-sec(A))=-2cot^(2)(A)

Answer 1

To verify this identity, we utilize Pythagorean identities and the reciprocal identities. The more you verify identities, the easier it gets. I hope I'm of help for you

Solution:

1. Let's first work on the left side of the equation. Make it only fraction. We would get something like (1-secx +1+secx)/((1-secx)(1+secx))
2. Simplify and we get 2/((1-secx)(1+secx))
3. 2/(1-sec^2(x)) *denominator is difference of squares*
4. Use Pythagorean identity: 1 + tan^2(x)=sec^2(x)
5. 1-sec^2(x)=-tan^2(x) *manipulated form of identity*
6. 2/(-tan^2(x))=-2cot^2(x) *replaced denominator with identity*
7. Use reciprocal identity: 1/tanx = cotx
8. -2cot^2(x)--2cot^2(x) *replaced 1/tan with cot*