Question : If $2\left (\cos^{2}\theta-\sin ^{2}\theta \right)=1$; ($\theta$ is a positive acute angle), then $\cot\theta$ is equal to:
Option 1: $-\sqrt3$
Option 2: $1\frac{1}{\sqrt3}$
Option 3: $1$
Option 4: $\sqrt3$
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Correct Answer: $\sqrt3$
Solution :
Given: $2\left (\cos^{2}\theta-\sin ^{2}\theta \right)=1$
⇒ $2\cos^{2}\theta-2\sin ^{2}\theta=\cos^{2}\theta+\sin ^{2}\theta$
⇒ $\cos^2\theta = 3\sin^2\theta$
⇒ $\frac{\cos^2\theta}{\sin^2\theta} = 3$
⇒ $\cot^2\theta=3$
⇒ $\cot\theta = \pm\sqrt3$
As $\theta$ is a positive acute angle,
So, $\cot\theta = \sqrt3$
Hence, the correct answer is $\sqrt3$.
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