If a = xcosθ + ysinθ and b = xsinθ − ycosθ, then a² + b² is equal to:

Question

Accepted Answer

Given:
 a = xcosθ + ysinθ
 b = xsinθ − ycosθ

We need to find a² + b².

Calculate a²:
 a² = (xcosθ + ysinθ)² = x²cos²θ + 2xycosθsinθ + y²sin²θ

Calculate b²:
 b² = (xsinθ − ycosθ)² = x²sin²θ − 2xysinθcosθ + y²cos²θ

Now, add a² and b²:
 a² + b² = (x²cos²θ + 2xycosθsinθ + y²sin²θ) + (x²sin²θ − 2xysinθcosθ + y²cos²θ)

Simplify:
 a² + b² = x²(cos²θ + sin²θ) + y²(sin²θ + cos²θ) + 2xycosθsinθ − 2xysinθcosθ

Note that 2xycosθsinθ − 2xysinθcosθ = 0
Also, cos²θ + sin²θ = 1

Therefore:
 a² + b² = x²(1) + y²(1) = x² + y²

Hence, the correct answer is x² + y².

Answer

Answer

Answer