Question

If $\sin A+{\sin }^{2}A=1$ Then,$({\cos }^{2}A+{\cos }^{4}A)=?$

Answer 1

Given: $\sin A+{\sin }^{2}A=1$

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

$\Rightarrow \sin A={\cos }^{2}A\dots \dots \left(i\right)$

$\text{Using this in above equation}$

$[{\sin }^{2}A+{\cos }^{2}A=1,\text{so,}{\cos }^{2}A=1-{\sin }^{2}A]$

Now, let's take, ${\cos }^{2}A+{\cos }^{4}A$

${\cos }^{2}A+{\cos }^{4}A$

$={\cos }^{2}A+({\cos }^{2}A{)}^2$

$=\sin A+{\sin }^{2}A[\text{Using (i)},\sin A={\cos }^{2}A]$

$=1[\text{Given,}\sin A+{\sin }^{2}A=1]$

$\therefore {\cos }^{2}A+{\cos }^{4}A=1$