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Inverse Trigonometric Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsInverse Trigonometric Functions1 Mark
Question
Solve the equation: $2\tan^{-1} (\cos x) = \tan^{-1} (2\  cosec\ x)$
✓

Answer

Here, we are required to find the value of $x,$
Now, the given equation is $ 2 \tan ^ { - 1 } ( \cos x ) = \tan ^ { - 1 } ( 2 cosec x )$
$ \Rightarrow$ $ \tan ^ { - 1 } \left( \frac { 2 \cos x } { 1 - \cos ^ { 2 } x } \right) = \tan ^ { - 1 } \left( \frac { 2 } { \sin x } \right)$ $ \left[ \because 2 \tan ^ { - 1 } x = \tan ^ { - 1 } \left( \frac { 2 x } { 1 - x ^ { 2 } } \right) ; - 1 < x < 1 \right]$
$ \Rightarrow$ $ \frac { 2 \cos x } { \sin ^ { 2 } x } = \frac { 2 } { \sin x }$ $ \left[ \because 1 - \cos ^ { 2 } x = \sin ^ { 2 } x \right] $
$ \Rightarrow \sin x \cos x-\sin ^2 x=0 $
$ \Rightarrow \sin x(\cos x-\sin x)=0 $
$ \Rightarrow \sin x=0 \text { or } \cos x=\sin x $
$ \Rightarrow \sin x=\sin 0 \text { or } \cot x=1=\cot \pi / 4$
$\therefore$ $x = 0 \text { or } \frac { \pi } { 4 }$
But here at $x = 0,$ the given equation does not exist.
Hence, $x = \frac { \pi } { 4 }$ is the only solution.

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