Write the given expression in terms of x and y only. cos(sin^-1(x) - tan^-1(y)).

Solution:

Given cos(sin^-1(x) - tan^-1(y)) --------(1)

Put sin^-1(x) = ⍺ and tan^-1(y) = β

cos ⍺ = √(1 - x²) and sin ⍺ = x 

cos β = 1/ √(1 + y²) and sin β = y/ √(1 + y²)

Substituting the terms in equation (1), cos(sin^-1(x) - tan^-1(y))

cos (⍺ - β) = cos ⍺ . cos β + sin ⍺ . sin β

cos (⍺ - β) = √(1 - x²) . [1/ √(1 + y²)] + x . [y/ √(1 + y²)]

cos (⍺ - β) = [√(1 - x²) + xy] / [√(1 + y²)]

Write the given expression in terms of x and y only. cos(sin^-1(x) - tan^-1(y)).

Summary:

Thee given expression in terms of x and y only for cos(sin^-1(x) - tan^-1(y)) is [√(1 - x²) + xy] / [√(1 + y²)]