Ask Me Help Desk - Trig verification step by step?

It took a while for me to understand what you're asking, but I believe it's this:

$ \cos(2y) = \frac {2 - \sec^2y} {\sec ^2y} $

The right hand side divides out like this:

$ \frac {2 - \sec^2y} {\sec ^2y} = \frac 2 {\sec^2y} - \frac {\sec^2y} {\sec ^2y} = \frac 2 {\sec^2y} -1 $

One of the first things to do when presented with functions of secant, cosecant, cotangent etc is to convert them to their sine and cosine equivalents - that makes it much easier to handle. So the right hand side becomes:

$ \frac 2 {\sec^2y} -1 = \frac 2 {1/\cos^2y} - 1 = 2 cos^2y -1 $

Now hopefully you can see how it equals the left hand side: cos(2y).