Question : If $\theta$ is a positive acute angle and $3(\sec^{2}\theta+\tan^{2}\theta)=5$, then the value of $\cos2\theta$ is:
Option 1: $\frac{1}{2}$
Option 2: $\frac{1}{\sqrt{2}}$
Option 3: $\frac{\sqrt{3}}{2}$
Option 4: $1$
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Correct Answer: $\frac{1}{2}$
Solution :
Given: $3(\sec^{2}\theta+\tan^{2}\theta)=5$
⇒ $3\sec^{2}\theta+3\tan^{2}\theta=5$
⇒ $6\tan^{2}\theta+3=5$
⇒ $\tan^{2}\theta=\frac{1}{3}$
We know that,
$\cos\theta=\frac{1- \tan^{2}\frac{\theta}{2}}{1+ \tan^{2}\frac{\theta}{2}}$
So, $\cos2\theta=\frac{1- \tan^{2}\theta}{1+ \tan^{2}\theta}=\frac{1-\frac{1}{3}}{1+\frac{1}{3}}=\frac{\frac{2}{3}}{\frac{4}{3}}={\frac{1}{2}}$
Hence, the correct answer is $\frac{1}{2}$.
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