Respuesta :

meerkat18
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:

[tex]sin(sin^{-1}(x) + cos^{-1}(y)) = x y + \sqrt{1 - y^2} \sqrt{1- x^2}[/tex]

Step-by-step explanation:

Given

[tex]sin(sin^{-1}(x) + cos^{-1}(y))[/tex]

Let's define

[tex]\alpha = sin^{-1}(x)[/tex]

[tex]\beta = cos^{-1}(y)[/tex]

Replacing

[tex]sin(\alpha + \beta)[/tex]

[tex]sin(\alpha) cos(\beta) + sin(\beta) cos(\alpha)[/tex]

But

[tex]sin(\alpha) = sin(sin^{-1}(x))=x[/tex]

[tex]cos(\beta) = cos(cos^{-1}(y))=y[/tex]

From trigonometric identity

[tex]sin^2(\beta) + cos^2(\beta) = 1[/tex]

[tex]sin(\beta) = \sqrt{1 - cos^2(\beta)} = \sqrt{1 - y^2}[/tex]

[tex]sin^2(\alpha) + cos^2(\alpha) = 1[/tex]

[tex]cos(\alpha) = \sqrt{1 - sin^2(\alpha)} = \sqrt{1- x^2}[/tex]

Replacing

[tex]sin(sin^{-1}(x) + cos^-1 (y)) = x y + \sqrt{1 - y^2} \sqrt{1- x^2}[/tex]