Question :
If $\tan2\theta\tan3\theta=1$ where $0°<\theta<90°$, then the value of $\theta$ is:
Option 1: $22\frac{1}{2}°$
Option 2: $18°$
Option 3: $24°$
Option 4: $30°$
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Correct Answer: $18°$
Solution :
Given: $\tan2\theta\tan3\theta=1$
⇒ $\tan3\theta=\frac{1}{\tan2\theta}$
⇒ $\tan3\theta=\cot2\theta$
⇒ $\tan3\theta=\tan(90°–2\theta)$
⇒ $3\theta=(90°-2\theta)$
⇒ $5\theta=90°$
$\therefore\theta=18°$
Hence, the correct answer is $18°$.
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