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

Solution:

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

Put sin^-1(x) = ⍺ and cos^-1(y) = ꞵ

⇒ x = sin ⍺ and y = cos ꞵ

From figure, tan ⍺ = x / √(1 - x²) and tan ꞵ = √(1 - y²)/ y

tan (⍺ + ꞵ) = [tan ⍺ + tan ꞵ] / [ 1 - (tan ⍺ × tan ꞵ)]

tan (⍺ + ꞵ) = [ x / √(1 - x²) + √(1 - y²) / y]/ [ 1 - ((x / √(1 - x²) × √(1 - y²) / y))]

tan (⍺ + ꞵ) = [(xy + √(1 - y²)√(1 - x²)) / √(1 - x²)y] / [(y√(1 - x²) - x√(1 - y²))/ √(1 - x²)y]

tan (⍺ + ꞵ) = (xy + √(1 - y²)√(1 - x²)) / (y√(1 - x²) - x√(1 - y²))

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

Summary:

The given expression tan(sin^-1(x) + cos^-1(y)) in terms of x and y is tan (⍺ + ꞵ) = (xy + √(1 - y²)√(1 - x²)) / (y√(1 - x²) - x√(1 - y²)).