My question is to make the right side equal the left side

What you wrote is incorrect. You wrote this:

\frac {\tan x}{sec x} -1 = secx + \frac 1 {tanx}


You must learn to use parentheses to avoid mistakes like that. I suspect what you really meant is this:

tanx/(secx -1) = (secx + 1)/tanx

Hint: start by cross-multiplying the fractions on the left and right hand sides, then replace the secx and tanx functions with their sine and cosine equivalents.