Question

If $a\cos \theta +b\sin \theta =m$ and $a\sin \theta -b\cos \theta =n$, then $a^2+b^2=$

Answer 1

$m^2=a^2{\cos }^2\theta +b^2{\sin }^2\theta +2ab\sin \theta \cos \theta n^2=b^2{\cos }^2\theta +a^2{\sin }^2\theta -2ab\sin \theta \cos \theta$

So, $m^2+n^2=a^2({\sin }^2\theta +{\cos }^2\theta )+b^2({\sin }^2\theta +{\cos }^2\theta )=a^2+b^2$