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Inverse Trigonometric Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsInverse Trigonometric Functions1 Mark
Question
Solve the equation: 2tan-1 (cos x) = tan-1 (2 cosec x)
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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$  sinx cosx - sin2x = 0
$\Rightarrow$ sinx(cosx - sinx) = 0
$\Rightarrow$ sinx = 0 or cosx = sinx​​​​
$\Rightarrow$ 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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