Find imaginary part of ${\sin ^{ - 1}}\left( {\cos ec\theta } \right)$
1. $\log \left[ {\cot \left( {\dfrac{\theta }{2}} \right)} \right]$
2. $\dfrac{\pi }{2}$
3. $\left( {\dfrac{1}{2}} \right)\log \left[ {\cot \left( {\dfrac{\theta }{2}} \right)} \right]$
4. None of these

Hint: Here in this question, we are given the function ${\sin ^{ - 1}}\left( {\cos ec\theta } \right)$ and we have to find the imaginary part of it. The first step is to let the given function be equal to the general form of a complex number. Take the inverse part to the other sides and apply the inverse trigonometric formula of a complex angle. Then compare the real and imaginary parts of both sides and you’ll get two equations from the equation of the imaginary part calculate the value of $x$ and put it in equation (1) of the real part. Now apply a trigonometric inverse formula to convert the function into a logarithm and solve further.

Formula Used:
Trigonometric Formula of complex angles –
$\sin \left( {x + iy} \right) = \sin x.\cosh y + i\cos x.\sinh y$
Trigonometric formula –
$cose{c^2}\theta - 1 = {\cot ^2}\theta $
Inverse formula –
${\cosh ^{ - 1}}x = \log \left( {x + \sqrt {{x^2} - 1} } \right)$

Complete step by step Solution:
Given that, to find the imaginary part of the function ${\sin ^{ - 1}}\left( {\cos ec\theta } \right)$
Let, ${\sin ^{ - 1}}\left( {cosec\theta } \right) = x + iy$
Therefore, $cosec\theta = \sin \left( {x + iy} \right)$
Using trigonometric formula,
$cosec\theta = \sin x.\cosh y + i\cos x.\sinh y$
Compare the real and imaginary part of both the sides,
$\sin x\cosh y = cosec\theta - - - - - \left( 1 \right)$
$\cos x.\sinh y = 0 - - - - - \left( 2 \right)$
From equation (2),
$\cos x = 0$
Which implies that,$x = \dfrac{\pi }{2}$
Now put the above value in equation (1),
$\sin \left( {\dfrac{\pi }{2}} \right)\cosh y = cosec\theta $
$1\left( {\cosh y} \right) = cosec\theta $
$y = {\cosh ^{ - 1}}\left( {cosec\theta } \right)$
Using trigonometric inverse formula,
$y = \log \left( {cosec\theta + \sqrt {cose{c^2}\theta - 1} } \right)$
Applying the trigonometric identity,
$y = \log \left( {cosec\theta + \cot \theta } \right)$
$y = \log \left( {\cot \dfrac{\theta }{2}} \right)$

Hence, the correct option is 1.

Note: The key concept involved in solving this problem is a good knowledge of complex numbers. Students must remember that complex numbers are made up of two parts: a real number and an imaginary number. Complex numbers serve as the foundation for more complex math, such as algebra. They have many practical applications, particularly in electronics and electromagnetism.