Question : $\sin ^4 \theta+\cos ^4 \theta$ in terms of $\sin \theta$ can be written as:
Option 1: $2 \sin ^4 \theta+2 \sin ^2 \theta-1$
Option 2: $2 \sin ^4 \theta-2 \sin ^2 \theta$
Option 3: $2 \sin ^4 \theta-2 \sin ^2 \theta-1$
Option 4: $2 \sin ^4 \theta-2 \sin ^2 \theta+1$
Correct Answer: $2 \sin ^4 \theta-2 \sin ^2 \theta+1$
Solution :
We know that, $\sin^2 \theta + \cos^2 \theta = 1$
⇒ $\cos^2 \theta = 1-\sin^2 \theta$
Now, $\sin^4 \theta + \cos^4 \theta$
$=(\sin^2 \theta + \cos^2 \theta)^2 - 2\sin^2 \theta \cos^2 \theta$
$=1^2 - 2\sin^2 \theta \cos^2 \theta$
$=1 - 2\sin^2 \theta (1 - \sin^2 \theta)$ [$\because\cos^2 \theta = 1-\sin^2 \theta$]
$= 1 - 2\sin^2 \theta + 2\sin^4 \theta$
Hence, the correct answer is $2 \sin ^4 \theta-2 \sin ^2 \theta+1$.
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