Question

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

Answer 1

I will use alpha in place of theta.

x²+y²

=(a sinα + b cosα)² + (a cosα -b sinα)²

=a² sin²α +2ab.sinα.cosα + b²cos²α + a²cos²α -2ab.cosα.sinα + b²sin²α

=a²sin²α + a²cos²α + b²cos²α + b²sin²α

=a²(cos²α+sin²α) + b²(cos²α + sin²α)

=a²+b²