prove identity

Question

(1/sinx + 1/cosx)/(1/sinx - 1/cosx) = (cos²x-sin²x)/1-2cosxsinx)

Answer 1

Start with the left-hand side and clear the fractions like so:

 

(1/sinx +1/cosx)/(1/sinx-1/cosx) = (cosx + sinx)/(cosx - sinx).

 

Then, multiply both the numerator and the denominator by cosx - sinx and we get:

 

(cos²x - sin²x)/(cos²x - 2cosxsinx + sin²x)

 

However, sin²x + cos²x = 1, and after substituting we get:

 

(cos²x - sin²x)/(1 - 2cosxsinx) and we're finished.

 

Just for fun, notice by using the double-angle identities for sin and cos, we can simplify further:

 

cos(2x)/(1 - sin(2x))