Question:
If $\sin \theta+\sin ^{2} \theta=1$, then $\cos ^{2} \theta+\cos ^{4} \theta=$
(a) −1
(b) 1
(c) 0
(d) None of these
Solution:
Given:
$\sin \theta+\sin ^{2} \theta=1$
$\Rightarrow 1-\sin ^{2} \theta=\sin \theta$
Now,
$\cos ^{2} \theta+\cos ^{4} \theta$
$=\cos ^{2} \theta+\cos ^{2} \theta \cos ^{2} \theta$
$=\cos ^{2} \theta+\left(1-\sin ^{2} \theta\right)\left(1-\sin ^{2} \theta\right)$
$=\cos ^{2} \theta+\sin \theta \sin \theta$
$=\cos ^{2} \theta+\sin ^{2} \theta$
$=1$
Hence, the correct option is (b).
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