Simplify, \[{\sin ^{ - 1}}(\cos x)\]

Hint: We have to simplify \[{\sin ^{ - 1}}(\cos x)\] to simplify this firstly we convert $\cos x$ into \[\sin \]function. We know that \[\cos \theta = \sin (90 - \theta )\] so we apply this identity on \[\cos x\]. This will convert \[\cos x\] into $\sin x$ putting the value of \[\cos x\] in \[{\sin ^{ - 1}}(\cos x)\] will give a simplified form of this function.

Complete step by step solution:
We have given that ${\sin ^{ - 1}}(\cos x) - - - - - - - - - - (i)$
We know that the value of $\cos \theta = \sin \left( {\dfrac{\pi }{2} - \theta } \right)$
So for$\cos x$, we have this
$\cos x = \sin \left( {\dfrac{\pi }{2} - x} \right)$
Putting in equation $(i)$
${\sin ^{ - 1}}(\cos x) = {\sin ^{ - 1}}\left( {\sin \left( {\dfrac{\pi }{2} - x} \right)} \right)$
=$\dfrac{\pi }{2} - x$
So we get ${\sin ^{ - 1}}(\cos x) = $$\dfrac{\pi }{2} - x$
The simplified form of ${\sin ^{ - 1}}(\cos x) = $$\dfrac{\pi }{2} - x$

Note: Trigonometry the branch of mathematics concerned with specific functions of angles and their applications to calculations. There are six trigonometric functions of the angle commonly used in trigonometry. Their names are sin($\sin $),cosine($\cos $),tangent($\tan $),cotangent ($\cot $),secant(sec) ,cosecant($\cos ec$).These trigonometric functions are related to the angle and the ratio of the sides of the triangle. Tangent is the ratio of the side opposite to the angle and the side adjacent to the angle. ‘sine’ is the ratio of the side opposite the angle and the hypotenuse. Inverse trigonometric functions are inverse of the trigonometric functions specifically they are inverse of sine, cosine, tangent, cotangent, secant, and cosecant functions. They are used to obtain an angle from any of the angles trigonometric ratio. The trigonometric function opposite operation of the trigonometric functions.