We can start with either side. It is always best to convert all trig functions given to sine and cosine when dealing with trig identities. I will begin on the right side.
I will also use x for theta. At the end, all you have to do is replace my x with your theta.
tan^2 x = sin^2x/cos^2x
sin^2 = 1 - cos^2x
[(sin^2x)/(cos^2x)]*[(1 - cos^2x)/1)] = LHS
Let LHS = left hand side
[sin^2x - sin^2x*cos^2x]/cos^2x = LHS
sin^2x(1 - cos^2x)/cos^2x = LHS
(sin^2x)/(cos^2x) - (1 - cos^2x)/1 = LHS
tan^2x - sin^2x = tan^2x - sin^2x