Question

Prove: 2 - sec^x / 1+ tan ^x = cos 2x

Answer 1

I assume you mean

(2-sec^2 x)/(1+tan^2 x)
since sec^2 x = 1+tan^2 x, we have

(1-tan^2 x)/(sec^2 x)
multiply top and bottom by cos^2 x
= (cos^2 x - sin^2 x)/1
= cos(2x)