Sin inverse x + sin inverse y + sin inverse z =πThen prove that x✓1-y square + y✓1-x square =z

To tackle the equation sin inverse x + sin inverse y + sin inverse z = π, we need to manipulate and analyze the relationship between the variables involved. Our objective is to prove the equation x√(1 - y²) + y√(1 - x²) = z. Let’s break this down step by step.

Understanding the Inverse Sine Function

The inverse sine function, denoted as sin inverse or arcsin, gives us an angle whose sine is a specific value. Since sin inverse x, sin inverse y, and sin inverse z are all angles, their sum being π indicates that they are related in such a way that they form a straight angle. This is crucial in our proof.

Using Trigonometric Identities

From the equation sin inverse x + sin inverse y + sin inverse z = π, we can isolate sin inverse z:

• sin inverse z = π - (sin inverse x + sin inverse y)

Now, by taking the sine of both sides, we use the sine addition formula:

• sin(π - θ) = sin θ

Thus, we have:

• z = sin(sin inverse x + sin inverse y)

Applying the Sine Addition Formula

Next, we apply the sine addition formula:

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

In our case, A = sin inverse x and B = sin inverse y:

• z = sin(sin inverse x) cos(sin inverse y) + cos(sin inverse x) sin(sin inverse y)
• z = x√(1 - y²) + √(1 - x²)y

Rearranging the Expression

Now we can rewrite the expression for z:

• z = x√(1 - y²) + y√(1 - x²)

This matches the form we need to prove, which is:

• x√(1 - y²) + y√(1 - x²) = z

Conclusion of the Proof

Since we derived the expression for z from the original equation and it matches the required form, we have effectively proven that if sin inverse x + sin inverse y + sin inverse z = π, then indeed x√(1 - y²) + y√(1 - x²) = z. This relationship highlights the interconnected nature of trigonometric functions and their inverses, demonstrating how angles and their sine values interact in a geometric context.