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

Solution:

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

We have to write the expression in terms of x and y.

Let sin^-1(x) = α

So, x = sin α

Let cos^-1(y) = β

So, y = cos β

From the figure, cos α = √(1 - x²)

From the figure, sin β = √(1 - y²)

So, sin(sin^-1(x) + cos^-1(y)) = sin(α + β)

By using trigonometric identities,

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

= xy + [(√(1 - x²))(√(1 - y²))]

Therefore, the expression in terms of x and y is xy + [(√(1 - x²))(√(1 - y²))].

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

Summary:

The expression sin(sin^-1x + cos^-1y) in terms of x and y only is xy + [(√(1 - x²))(√(1 - y²))].