Question : If $\theta$ is a positive acute angle and $\tan2\theta \tan3\theta=1$, then the value of $2(\cos^2\frac{5\theta}{2}-1)$ is:
Option 1: $-\frac{1}{2}$
Option 2: -1
Option 3: 0
Option 4: $\frac{1}{2}$
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Correct Answer: 0
Solution :
Given:
$\tan2\theta \tan3\theta=1$
or, $\tan2\theta=\cot3\theta$
or, $\tan2\theta=\tan(90-3\theta)$
or, $2\theta=90-3\theta$
or, $5\theta =90$
$\therefore \theta=18°$
$2(\cos^2\frac{5\theta}{2}-1)$
= $2(\cos^2\frac{5×18°}{2}-1)$
= $2(\cos^245°-1)$
= $2×((\frac{1}{\sqrt{2}})^2-1)$
= $2×-\frac{1}{2}$
= $-1$
Hence, the answer is -1.
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