If $cos x + {cos}^{2} x = 1$ , prove that ${sin}^{2} x + {sin}^{4} x = 1$ .

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Solution

Given, $c o s x + c o s^{2} x = 1$
$\Rightarrow c o s x = 1 - c o s^{2} x$
$\Rightarrow c o s x = s i n^{2} x$
Squaring both sides we get
$\Rightarrow c o s^{2} x = s i n^{4} x$ .....(1)
$\Rightarrow s i n^{4} x - c o s^{2} x = 0$
Adding both side 1, we get
$\Rightarrow s i n^{4} x + 1 - c o s^{2} x = 1$
$∴ s i n^{2} x + s i n^{4} x = 1$ Proved.

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