Question

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

Answer 1

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))