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What is the sin^{-1 }x + sin^{-1 }y formula?

Trigonometry is the branch of mathematics that deals with relationships of angles and sides of a triangle. Inverse trigonometric functions are the inverse ratio of the basic trigonometric ratios. They have immense applications in calculus. Let's look into a concept related to inverse trigonometric functions.

## Answer: The formula for sin^{-1 }x + sin^{-1 }y is sin^{-1 }[x√ (1 - y^{2}) + y√ (1 - x^{2})]

Let's understand the solution in detail.

**Explanation:**

Let's derive the formula for the given expression.

Let sin^{-1 }x = A, hence sin A = x.

Hence, we can see that cos A = √ (1 - x^{2}).

Let sin^{-1 }y = B, hence sin B = y.

Hence, we can see that cos B = √ (1 - y^{2}).

Now, we know that:

sin (A + B) = sin A.cos B + cos A.sin B

Now, substituting the values in the above equation:

sin (A + B) = x√(1 - y^{2}) + y√(1 - x^{2})

⇒ A + B = sin^{-1 }[x√(1 - y^{2}) + y√(1 - x^{2})]

⇒sin^{-1 }x + sin^{-1 }y = sin^{-1 }[x√(1 - y^{2}) + y√(1 - x^{2})].