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Inverse Trigonometric Functions — MATHS STD 12 Science — Question

Rajasthan 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^{2x} = 0$
$ \Rightarrow \sin x\ (\cos\ x - \sin\ x) = 0$
$ \Rightarrow \sin x = 0 or \cos x = \sin\ x​​​​$
$ \Rightarrow \sin x = \sin 0$ 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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