Question : If $\sin \theta \cos \theta=\frac{1}{\sqrt{3}}$ then the value of $\left(\sin ^4 \theta+\cos ^4 \theta\right)$ is:
Option 1: $1$
Option 2: $\frac{5}{3}$
Option 3: $\frac{2}{3}$
Option 4: $\frac{1}{3}$
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Correct Answer: $\frac{1}{3}$
Solution :
Given: $\sin \theta \cos \theta=\frac{1}{\sqrt{3}}$
We know that $\sin^2 \theta + \cos^2 \theta = 1$
Squaring both sides,
$⇒\sin^4 \theta + \cos^4 \theta + 2\sin^2\theta\cos^2\theta= 1$
$⇒\sin^4 \theta + \cos^4 \theta + 2(\sin\theta\cos\theta)^2 = 1$
$⇒\sin^4 \theta + \cos^4 \theta + 2\times (\frac{1}{\sqrt{3}})^2 = 1$
$⇒\sin^4 \theta + \cos^4 \theta + \frac{2}{3} = 1$
$⇒\sin^4 \theta + \cos^4 \theta = 1 - \frac{2}{3} = \frac{1}{3}$
Hence, the correct answer is $\frac{1}{3}$.
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