Trigonometric Identities and Equations

Question

If x=a sin  θ - b cos  θ  and y=a cos  θ +b sin  θ , show that x^2+y^2=a^2+b^2

Answer 1

x = a sinθ - b cosθ

y = a cosθ - b sinθ

 

x² + y² = a² + b²

 

x² = a²sin²θ - 2absinθcosθ + b²cos²θ

y² = a²cos²θ + 2absinθcosθ + b²sin²θ

 

x² + y² = a²sin²θ - 2absinθcosθ + b²cos²θ + a²cos²θ + 2absinθcosθ + b²sin²θ

x² + y² = a²sin²θ + b²cos²θ + a²cos²θ + b²sin²θ

 

rearranging terms

x² + y² = a²(sin²θ + cos²θ) + b²(sin²θ + cos²θ)

Recall that sin²θ + cos²θ = 1

Therefore

x² + y² = a² + b²