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Find the the value of \theta if \tan ^{-1}(\frac{1+\sin x}{\cos x})=2x

Option: 1

\frac{\pi}{2}


Option: 2

\frac{\pi}{3}


Option: 3

\frac{\pi}{4}


Option: 4

\frac{\pi}{6}


Answers (1)

best_answer

 

 

Graph of Principal Value of function f-1 (f (x)) (Part 2) -

Graph of Principal Value of function f-1 (f (x)) (Part 2)

  1. Graph of tan-1( tan (x))

The domain of the function is R and range is (- π/2, π/2)

            y = tan-1( tan (x))

∴          tan y = tan x

⇒         y = nπ + x, n ∈ I (integer)  

 

To draw the graph of y = tan-1( tan (x)), we draw all the line of y = nπ + x, n ∈ I for 

y∈(- π/2, π/2).

 


 

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\tan ^{-1}(\frac{1+\sin x}{\cos x})=2x\\ \begin{array}{l}{\tan ^{-1}\left[\frac{1+\sin x}{\cos x}\right]=\tan ^{-1}\left[\frac{1+\cos (\pi / 2-x)}{\sin (\pi / 2-x)}\right]} \\ {=\tan ^{-1}\left[\frac{2 \cos ^{2}(\pi / 4-x / 2)}{2 \sin (\pi / 4-x / 2)}\right]=\tan ^{-1} \cot \left(\frac{\pi}{4}-\frac{x}{2}\right) =\frac{\pi}{4}+\frac{x}{2}}\end{array}\\ \frac{\pi}{4}+\frac{x}{2}=2x \Rightarrow x=\frac{\pi}{6}

Posted by

Pankaj Sanodiya

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