Question

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

Answer 1

we are given the expression sin (sin-1 (X) + cos-1 (y)). We use the associative property to distribute the sine function. sin (sin-1 (x)) is equal to x. cos-1 (y) is equal to beta, the other angle besides alpha. sin beta is also equal to x which means the simplified term is 2x. This makes sense because sin-1 x = cos-1 y 

Answer 2

`sin(sin^(-1)(x) + cos^(-1)(y)) = x y + √(1 - y^2) √(1- x^2)`

Explanation:

Given

`sin(sin^(-1)(x) + cos^(-1)(y))`

Let's define

`\alpha = sin^(-1)(x)`

`\beta = cos^(-1)(y)`

Replacing

`sin(\alpha + \beta)`

`sin(\alpha) cos(\beta) + sin(\beta) cos(\alpha)`

But

`sin(\alpha) = sin(sin^(-1)(x))=x`

`cos(\beta) = cos(cos^(-1)(y))=y`

From trigonometric identity

`sin^2(\beta) + cos^2(\beta) = 1`

`sin(\beta) = √(1 - cos^2(\beta)) = √(1 - y^2)`

`sin^2(\alpha) + cos^2(\alpha) = 1`

`cos(\alpha) = √(1 - sin^2(\alpha)) = √(1- x^2)`

Replacing

`sin(sin^(-1)(x) + cos^-1 (y)) = x y + √(1 - y^2) √(1- x^2)`