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Question

The number of solutions of $$\tan x+\sec x= 2\cos x$$ in $$\left [ 0,2\pi  \right ]$$ is


A
2
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B
3
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C
0
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D
1
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Solution

The correct option is B 3
$$\tan x+\sec x= 2 \cos x$$
$$\Rightarrow 1+\sin x= 2\cos^{2}x $$
$$\Rightarrow 1+ \sin x = 2\left ( 1- \sin^{2}x \right )$$
$$\Rightarrow 2 \sin^{2}x +\sin x-1= 0 $$
$$\Rightarrow \left ( 2\sin x- 1 \right ) \left ( 1+\sin x \right )= 0 $$
$$\Rightarrow \sin x= \displaystyle \frac{1}{2}, \sin x= -1$$

So there are three solutions like $$x=30^{0}, 150^{0}, 270^{0}$$

190977_161468_ans.png

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