Write following expression as an algebraic expression contai

Write following expression as an algebraic expression containing U & V cos(tan^-1 U + tan^-1 V)

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

let tan^-1 u = a

so , tan a = u

sin a = u / sqrt ( u^2 + 1 )

cos a = 1 / sqrt ( u^2 + 1)

tan^-1 v = b

tan b = v

sin b = v / sqrt ( v^2 + 1 )

cos b = 1 / sqrt (v^2 +1 )

so we get

cos (a + b )

applying cos (a+b) formula

cos (a + b) = cos a cos b - sin a sin b

plugging the values we get

cos (a+b) = 1 / sqrt ( u^2 + 1) * 1 / sqrt (v^2 +1 ) - u / sqrt ( u^2 + 1)* v/ sqrt ( v^2 + 1)

= 1 / sqrt sqrt (u^2 +1 )* sqrt (v^2 +1 )- uv / sqrt (u^2 +1 )* sqrt (v^2 +1 )

cos ( tan^-1 u + tan^-1 v ) = ( 1 - uv) / sqrt (u^2 +1 )* sqrt (v^2 +1 )