Question

The number of solutions of $\tan \left(\frac{\pi }{2}\sin \theta \right)=\cot \left(\frac{\pi }{2}\cos \theta \right),\theta ϵ[0,6\pi ],$ is

• A. $5$
• B. $7$
• C. $4$
• D. $6$

Answer 1

The correct option is B $7$

$\tan \left(\frac{\pi }{2}\sin \theta \right)=\cot \left(\frac{\pi }{2}\cos \theta \right),\theta ϵ[0,6\pi ],$
$\tan \left(\frac{\pi }{2}\sin \theta \right)=\tan (\frac{\pi }{2}-\frac{\pi }{2}\cos \theta )$
$\frac{\pi }{2}\sin \theta =\frac{\pi }{2}-\frac{\pi }{2}\cos \theta$
$\sin \theta +\cos \theta =1$
$\Rightarrow 1+\sin 2\theta =1$
$\Rightarrow \sin 2\theta =0$
$\Rightarrow \theta =0,\pi ,2\pi ,3\pi ,4\pi ,5\pi ,6\pi$
Hence, total number of solutions is $7$