Question : If $2(\cos^{2}\theta-\sin^{2}\theta)=1$; ($\theta$ is a positive acute angle), then $\cot\theta$ is equal to:
Option 1: $–\sqrt{3}$
Option 2: $\frac{1}{\sqrt{3}}$
Option 3: $1$
Option 4: $\sqrt{3}$
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Correct Answer: $\sqrt{3}$
Solution :
Given: $2(\cos^{2}\theta-\sin^{2}\theta)=1$
⇒ $\cos^{2}\theta-(1-\cos^{2}\theta)=\frac{1}{2}$
⇒ $2\cos^{2}\theta=1+\frac{1}{2}$
⇒ $\cos^{2}\theta=\frac{3}{2×2}$
⇒ $\cos\theta=\frac{\sqrt{3}}{2}$
We know that [$\cos30°=\frac{\sqrt{3}}{2}$]
⇒ $\cos\theta=\cos30°$
$\therefore\theta=30°$
Then, $\cot30°=\sqrt{3}$
Hence, the correct answer is $\sqrt{3}$.
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