In $(0,2 \pi)$, the number of solutions of $ an heta+\sec heta=2 \cos heta$ are

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In $(0,2 \pi)$, the number of solutions of $	an 	heta+\sec 	heta=2 \cos 	heta$ are

Aicharya · Accepted Answer

$\begin{aligned} & 	an 	heta+\sec 	heta=2 \cos 	heta \ & \Rightarrow \frac{\sin 	heta}{\cos 	heta}+\frac{1}{\cos 	heta}=2 \cos 	heta \ & \Rightarrow \sin 	heta+1=2 \cos ^2 	heta \ & \Rightarrow \sin 	heta+1=2\left(1-\sin ^2 	hetaight) \ & \Rightarrow 2 \sin ^2 	heta+\sin 	heta-1=0\end{aligned}$ $\begin{aligned} 	herefore \quad \sin 	heta= & -1, \frac{1}{2} \ 	ext { For } \sin 	heta & =-1, \sin 	heta=\frac{1}{2} \ 	heta & =\frac{3 \pi}{2} \quad 	heta=\frac{\pi}{6}, \frac{5 \pi}{6} \end{aligned}$ $	herefore \quad$ Number of solutions $=3$

In $(0,2 \pi)$, the number of solutions of $\tan \theta+\sec \theta=2 \cos \theta$ are

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