Question

If x = a sin θ + b cos θ and y = a cos θ - b sin θ, prove that X 2 + y 2 = a 2 + b 2

Answer 1

x=asinθ+bcosθ
∴, x²=(asinθ+bcosθ)²
       =a²sin²θ+2absinθcosθ+b²cos²θ
y=acosθ-bsinθ
∴, y²=(acosθ-bsinθ)²
        =a²cos²θ-2absinθcosθ+b²sin²θ
∴, x²+y²=a²sin²θ+2absinθcosθ+b²cos²θ+a²cos²θ-2absinθcosθ+b²sin²θ
             =a²(sin²θ+cos²θ)+b²(sin²θ+cos²θ)
             =a²+b² (Proved) [∵, sin²θ+cos²θ²=1]