Question

If x = a cos theta - b sin theta y = a sin theta +b cos theta, then prove that a 2 + b 2 = x 2 + y 2

Answer 1

x = acosθ - bsinθ
y = asinθ +bcosθ

squring both sides and adding,

x²+y² = (acosθ - bsinθ)² + (asinθ +bcosθ)²
         =a²cos²θ +b²sin²θ + 2abcosθ sinθ + a²sin²θ + b²cos²θ + 2abcosθ sinθ
         =a²(cos²θ + sin²θ) +b²(sin²θ +cos²θ)
         =a² +b²
         = LHS