The principal solutions of the equation x+ x=2 x are

Correct Answer is : π6, 5π6π/6,\ 5π/66π, 65π

Correct Answer is : π6, 5π6π/6,\ 5π/66π, 65π

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MHTCET 14 May 2023 Shift 1 Solved Paper

Section: Mathematics

Question : 75 of 150

Marks: +1, -0

\sec x+\tan x=2 \cos x

are

Solution:

The given equation is defined for

x \neq \frac{\pi}{2},\ \frac{3\pi}{2}

Now,

\tan x+\sec x=2 \cos x

\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 (1-\sin^2 x)

\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

\left\{ \left[ \begin{array}{l} \because \sin x \neq -1 \text{ otherwise } \cos x=0 \text{ and } \\ \tan x\ \sec x \text{ will be undefined } \end{array} \right] \right\}

\Rightarrow \sin x = \frac{1}{2}

\Rightarrow x = \frac{\pi}{6},\ \frac{5\pi}{6} \text{ in } (0,2\pi)

\;

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