How do I write the expression cos(cos^-1(u)+sin^-1(v)) as an algebraic expression containing only u and v?

Question

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

Answer 1

arccos is defined where sin is greater than or equal to 0.

arcsin is defined where cos is greater than or equal to 0.

When cos = u, sin = √(1 - u²)

When sin = v, cos = √(1 -v²)

Then, as cos(a + b) = cos a cos b - sin a sin b,

cos(cos^-1(u)+sin^-1(v)) =

u√(1 -v²) - v√(1 - u²)