Question : The greatest value of $\sin ^{4 }\theta +\cos ^{4}\theta$ is:
Option 1: $2$
Option 2: $3$
Option 3: $\frac{1}{2}$
Option 4: $1$
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Correct Answer: $1$
Solution :
To find the maximum value of $\sin^4\theta +\cos^4\theta$
We can write this expression as,
$\sin^4\theta +\cos^4\theta=(\sin^2\theta)^2 +(\cos^2\theta)^2 +2\sin^2\theta \cos^2\theta- 2\sin^2\theta \cos^2\theta$
$=(\sin^2\theta+\cos^2\theta)^2- 2\sin^2\theta \cos^2\theta = 1-2\sin^2\theta \cos^2\theta$
$= 1-\frac{(\sin^22\theta)}{2}$
For the maximum value of the above function, $\frac{(\sin^22\theta)}{2}$ should have minimum value, i.e., 0.
So, Maximum Value = 1
Hence, the correct answer is $1$.
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