The principal solutions of the equation $\sec x+ an x=2 \cos x$ are

Question

The principal solutions of the equation $\sec x+	an x=2 \cos x$ are

Aicharya · Accepted Answer

The given equation is defined for $x 
eq \frac{\pi}{2}, \frac{3 \pi}{2}$. Now, $	an x+\sec x=2 \cos x$ $\begin{aligned} & \Rightarrow \frac{\sin x}{\cos x}+\frac{1}{\cos x}=2 \cos x \ & \Rightarrow(\sin x+1)=2 \cos ^2 x \ & \Rightarrow(\sin x+1)=2\left(1-\sin ^2 xight) \ & \Rightarrow(\sin x+1)=2(1-\sin x)(1+\sin x) \ & \Rightarrow(1+\sin x)[2(1-\sin x)-1]=0 \ & \Rightarrow 2(1-\sin x)-1=0 \ & \quad \ldots\left[\begin{array}{r} \sin x 
eq-1 	ext { otherwise } \cos x=0 	ext { and } \ 	an x, \sec x 	ext { will be undefined } \end{array}ight] \ & \Rightarrow \sin x=\frac{1}{2} \quad \ & \Rightarrow x=\frac{\pi}{6}, \frac{5 \pi}{6} 	ext { in }(0,2 \pi) \end{aligned}$

The principal solutions of the equation $\sec x+\tan x=2 \cos x$ are

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