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Question

$$2 tan^{-1} (\cos x)=tan^{-1}( 2 cosecx)$$ then $$x=?$$


A
π/4
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B
π/6
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C
π/2
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D
no solution
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Solution

The correct option is D $$\pi/4$$
Let $$k=2 tan ^{-1}(cos x)=tan ^{-1}(2 cosec x)$$
$$\displaystyle tan \frac{k}{2}=cos x$$ and $$tan k=2 cosec x$$
$$\displaystyle tan k=\frac{2 tan \frac{k}{2}}{1-tan ^{2}\frac{k}{2}}=\frac{2\times cos x}{1- cos ^{2}x}=\frac{2 cos x}{sin ^{2}x}=\frac{2}{sin x}$$
$$\Rightarrow tan x=1$$
$$\displaystyle \Rightarrow x=\frac{\pi }{4}$$

Mathematics

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