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Question: Find the value of given inverse trigonometric equation \({\sin ^{ - 1}}\left( {\cos \left( {{{\sin }...

Find the value of given inverse trigonometric equation sin⁡−1(cos⁡(sin⁡−1x))+cos⁡−1(sin⁡(cos⁡−1x)){\sin ^{ - 1}}\left( {\cos \left( {{{\sin }^{ - 1}}{\text{x}}} \right)} \right) + {\cos ^{ - 1}}\left( {\sin \left( {{{\cos }^{ - 1}}{\text{x}}} \right)} \right).

Explanation

Solution

Hint: We need to inverse trigonometric identities in this problem. Some of these identities are-
sin⁡−1x+cos⁡−1x=π2{\sin ^{ - 1}}{\text{x}} + {\cos ^{ - 1}}{\text{x}} = \dfrac{{\pi}}{2}
sinx=cos⁡(π2−x)sinx = \cos \left( {\dfrac{{\pi}}{2} - {\text{x}}} \right)

Complete step-by-step solution -
We have been given sin⁡−1(cos⁡(sin⁡−1x))+cos⁡−1(sin⁡(cos⁡−1x)){\sin ^{ - 1}}\left( {\cos \left( {{{\sin }^{ - 1}}{\text{x}}} \right)} \right) + {\cos ^{ - 1}}\left( {\sin \left( {{{\cos }^{ - 1}}{\text{x}}} \right)} \right). In order to simplify this, we will use the given identities-
sin⁡−1x+cos⁡−1x=π2{\sin ^{ - 1}}{\text{x}} + {\cos ^{ - 1}}{\text{x}} = \dfrac{{\pi}}{2}
sin⁡−1x=π2−cos⁡−1x....(1){\sin ^{ - 1}}x = \dfrac{{\pi}}{2} - {\cos ^{ - 1}}x....\left( 1 \right)
cos⁡−1x=π2−sin⁡−1x....(2){\cos ^{ - 1}}x = \dfrac{{\pi}}{2} - {\sin ^{ - 1}}x....\left( 2 \right)
Using the given equations (1) and (2) we can write the expression as-
=π2−cos⁡−1(cos⁡(sin⁡−1x))+π2−sin⁡−1(sin⁡(cos⁡−1x))= \dfrac{{\pi}}{2} - {\cos ^{ - 1}}\left( {\cos \left( {{{\sin }^{ - 1}}{\text{x}}} \right)} \right) + \dfrac{{\pi}}{2} - {\sin ^{ - 1}}\left( {\sin \left( {{{\cos }^{ - 1}}{\text{x}}} \right)} \right)
As we know that sin⁡−1(sinx)=x  and  cos⁡−1(cosx)=x{\sin ^{ - 1}}\left( {sinx} \right) = {\text{x}}\;\text{and}\;{\cos ^{ - 1}}\left( {cosx} \right) = {\text{x}}. So,
=π2−sin⁡−1x+π2−cos⁡−1x= \dfrac{{\pi}}{2} - {\sin ^{ - 1}}{\text{x}} + \dfrac{{\pi}}{2} - {\cos ^{ - 1}}{\text{x}}
=π−(sin⁡−1x+cos⁡−1x)= {\pi} - \left( {{{\sin }^{ - 1}}{\text{x}} + {{\cos }^{ - 1}}{\text{x}}} \right)
=π−π2=π2= {\pi} - \dfrac{{\pi}}{2} = \dfrac{{\pi}}{2}
This is the required answer.

Note: It is important to note that these formulas are valid only when the value of x is between 0 and 90o90^o. So, we used an assumption that all the angles are acute angles. If it is not mentioned what type of angle is it, then we assume the angles to be acute.