Question

If $\tan 7\theta .\tan 3\theta =1$, then find the value of $\theta$.

Answer 1

Consider the given equation.

$\tan 7\theta \cdot \tan 3\theta =1$

$\frac{\sin 7\theta }{\cos 7\theta }\cdot \frac{\sin 3\theta }{\cos 3\theta }=1$

$\sin 7\theta \cdot \sin 3\theta =\cos 7\theta \cdot \cos 3\theta$

$\cos 7\theta \cdot \cos 3\theta -\sin 7\theta \cdot \sin 3\theta =0$

We know that

$\cos (A+B)=\cos A\cos B-\sin A\sin B$

So,

$\cos (7\theta +3\theta )=0$

$\cos \left(10\theta \right)=0$

$\cos \left(10\theta \right)=\cos \frac{\pi }{2}$

$10\theta =2n\pi \pm \frac{\pi }{2}$

$\theta =\frac{n\pi }{5}\pm \frac{\pi }{20}$

Hence, the value of $\theta$ is $\frac{n\pi }{5}\pm \frac{\pi }{20}$.