Question

If $\left(a\sin A\right)=b\cos A$ and $a{\sin }^{3}A+b{\cos }^{3}A=\sin A\cos A$, then prove $a^2+b^2=1$

Answer 1

$a\sin A\left({\sin }^{2}A\right)+b\cos A\left({\cos }^{2}A\right)\sin A\cos A$

$a\sin A\left(1\right)=\sin A\cos A$

$\Rightarrow a=\cos A$

$\Rightarrow b=\sin A$

$\Rightarrow a^2+b^2={\sin }^{2}A+{\cos }^{2}A=1$.