Prove an Identity help!

Prove this identity:

cos^2y - sin^2y = (1-tan^2y) / 1 + tan^2y

Please show all steps as clearly as possible as i would like to understand what exactly is going on - comments throughout the solution explaining what is going on would be greatly appreciated!!

RHS = (1-tan²y)/(1+tan²y)

Multiply numerator and denominator by cos²y
(note that tan²ycos²y = sin²y)

RHS = (1*cos²y-tan²ycos²y)/(1*cos²y+tan²ycos²y…‡
(cos²y - sin²y)/(cos²y +sin²y)

Since cos²y + sin²y =1

RHS = (cos²y - sin²y)/1 = LHS