Question

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

Answer 1

let theta be o

x^2=a^2 cos^2 o + b^2 sin^2 o -2ab cos o sin o

y^2=a^2 sin^2 o + b^2 cos^2 o + 2 ab sin o cos o

then,
x^2 + y^2

= a^2(sin^2 o +cos^2 o) + b^2(sin^2 o + cos^2 o)

=a^2 + b^2