Question
If $\sin X+\sin ^2 X=1,$ prove that $\cos ^2 X+\cos ^4 X=1$.
✓
Answer
Given $\sin X+\sin ^2 X=1......... (i)$
$\Rightarrow \sin X=1-\sin ^2 X=\cos ^2 X .........(ii)$
Now we show that $\cos ^2 X+\cos ^4 X=1$
$\text { L.H.S }=\cos ^2 X+\cos ^4 X$
$=1-\sin ^2 X+\left(1-\sin ^2 X\right)^2 \text { [Using (ii)] }$
$=\sin X+\sin ^2 X[\text { Using (ii)] }$
$=1[\text { Using (i)] }$
$=\text { R.H.S }$
$\Rightarrow \sin X=1-\sin ^2 X=\cos ^2 X .........(ii)$
Now we show that $\cos ^2 X+\cos ^4 X=1$
$\text { L.H.S }=\cos ^2 X+\cos ^4 X$
$=1-\sin ^2 X+\left(1-\sin ^2 X\right)^2 \text { [Using (ii)] }$
$=\sin X+\sin ^2 X[\text { Using (ii)] }$
$=1[\text { Using (i)] }$
$=\text { R.H.S }$
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