Question

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

Answer 1

Answer:

$sin(sin^{-1}(x) + cos^{-1}(y)) = x y + \sqrt{1 - y^2} \sqrt{1- x^2}$

Step-by-step 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

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

$sin(\beta) = \sqrt{1 - cos^2(\beta)} = \sqrt{1 - y^2}$

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

$cos(\alpha) = \sqrt{1 - sin^2(\alpha)} = \sqrt{1- x^2}$

Replacing

$sin(sin^{-1}(x) + cos^-1 (y)) = x y + \sqrt{1 - y^2} \sqrt{1- x^2}$

Answer 2

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