Express sin (tan^-1 (u) - tan^-1 (v)) algebraically in terms of u and v

Let tan^-1 (u) = A

tan A = u, opposite side, u, adjacent side = 1, hypotenuse = √ (u^2+1)

sin A = u/√ (u^2+1)

cos A = 1/√ (u^2+1)

let tan^-1 (v) = B

tan B = v

sin B = v/√ (v^2+1)

cos B = 1/√ (v^2+1)

cos [ tan^-1 (u) + tan^-1 (v) ]

= cos (A + B)

= cos A cos B - sin A sin B

= (1/√ (u^2+1)) (1/√ (v^2+1)) - (u/√ (u^2+1)) (v/√ (v^2+1))

= (1 - uv) / √ (u^2+1)) √ (v^2+1))