Question : If $\sin \theta +\sin ^{2}\theta =1$, then the value of $\cos ^{2}\theta +\cos ^{4}\theta$ is:
Option 1: 2
Option 2: 4
Option 3: 0
Option 4: 1
Correct Answer: 1
Solution :
Given: $\sin \theta +\sin ^{2}\theta =1$
We know that, $\sin ^{2}\theta+\cos ^{2}\theta=1$
⇒ $\sin \theta=1-\sin ^{2}\theta=\cos ^{2}\theta$
To find: $\cos ^{2}\theta +\cos ^{4}\theta$
$=\cos ^{2}\theta(1 +\cos ^{2}\theta)$
Putting the values, we get:
$=\sin\theta(1 +\sin\theta)$
$\sin \theta +\sin ^{2}\theta = 1$
Hence, the correct answer is 1.
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