Question

Simplify the given expression $\sin ({\sin ^{ - 1}}x + {\cos ^{ - 1}}x)$?

Answer 1

Hint: In the given problem, we are required to calculate sine of a compound angle that is the sum of two angles; one whose sine is given to us as x and other whose cosine is given to us as x. Such problems require basic knowledge of trigonometric ratios and formulae. Besides this, knowledge of concepts of inverse trigonometry is extremely essential to answer these questions correctly.

Complete step by step answer:
In the given problem, we are required to find the sine of a compound angle which is given to us as a sum of two angles such that sine of one angle is given as x and cosine of other angle is given to us as x.
As we know the sine ratio of a given angle and cosine ratio of the given angle are complementary. Hence, if sine and cosine ratios are the same for two angles, then the angles must be complimentary.
Therefore, ${\sin ^{ - 1}}x + {\cos ^{ - 1}}x = \dfrac{\pi }{2}$ .
Now, taking sine on both sides of the equation given above, we get,
$\sin ({\sin ^{ - 1}}x + {\cos ^{ - 1}}x)$$ = \sin \left( {\dfrac{\pi }{2}} \right)$
$ \Rightarrow \sin ({\sin ^{ - 1}}x + {\cos ^{ - 1}}x) = 1$

Note: The given problem can also be solved by using the compound angle formula of sine $\sin (A + B) = \sin A\cos B + \cos A\sin B$. Then, we have to find sin of an angle whose cosine is given to us and vice versa. For finding a trigonometric ratio for an angle given in terms of an inverse trigonometric ratio, we have to first assume that angle to be some unknown, let's say $\theta $. Then proceeding further, we have to find the value of a trigonometric function of that unknown angle $\theta $. Then we find the required trigonometric ratio with help of basic trigonometric formulae and definitions of trigonometric ratios. Such questions require clarity of basic concepts of trigonometric functions as well as their inverse.