Question

Prove the identity
1-tan(x)tan(y)) = (cos(x+y)/cos(x)cos(y)

Answer 1

What is cos(x+y) equal to? I.e., expand it out.

Answer 2

Using the formula cos(A+B)=cos A cos B - sin A sin B

cos (x + y) cos(x)cos(y) - sin(x)sin(y) cos(x)cos(y) sin(x)sin(y)
RHS = ---------- = ---------------------- = ------------ - ------------
cos(x)cos(y) cos(x)cos(y) cos(x)cos(y) cos(x)cos(y)

sin(x) sin(y)
= 1 - ------- * ------ = 1 - tan(x)*tan(y) = 1 - tan(x)tan(y) = LHS
cos(x) cos(y)