Question

If $\sin A+{\sin }^{2}A=1$, then show that ${\cos }^{2}A+{\cos }^{4}A=1$.

Answer 1

We have, $\sin A+{\sin }^{2}A=1$

$\Rightarrow \sin A=1-{\sin }^{2}A$

$\Rightarrow (\sin A{)}^2=({\cos }^{2}A{)}^2$ [$\because {\sin }^{2}A+{\cos }^{2}A=1$]

$\Rightarrow {\sin }^{2}A={\cos }^{4}A$

$\Rightarrow 1-{\cos }^{2}A={\cos }^{4}A$

$\Rightarrow {\cos }^{2}A+{\cos }^{4}A=1$ [proved]