Solveeit Logo
Login

Question

Question: How do you simplify \[\cos \left( {{{\sin }^{ - 1}}x} \right)\] ?...

How do you simplify cos⁡(sin⁡−1x)\cos \left( {{{\sin }^{ - 1}}x} \right) ?

Explanation

Solution

Here, we will assume the inverse sine function to be some angle. Then using this we will find the value of xx and substitute it in the given expression. We will then simplify the expression using the properties of inverse trigonometric functions. We will then use the Pythagorean identity to simplify the equation and get the required value.

Formula Used:
Pythagorean identity : sin⁡2θ+cos⁡2θ=1{\sin ^2}\theta + {\cos ^2}\theta = 1

Complete Step by Step Solution:
We are given that cos⁡(sin⁡−1x)\cos \left( {{{\sin }^{ - 1}}x} \right).
Let us consider θ=sin⁡−1(x)\theta = {\sin ^{ - 1}}\left( x \right).
Thus, by rewriting the equation, we get
x=sin⁡θx = \sin \theta …………………………………………………….(1)\left( 1 \right)
Thus, by substituting the equation (1)\left( 1 \right) in the given expression, we get
cos⁡(sin⁡−1x)=cos⁡(sin⁡−1(sin⁡θ))\cos \left( {{{\sin }^{ - 1}}x} \right) = \cos \left( {{{\sin }^{ - 1}}\left( {\sin \theta } \right)} \right)
Now we know that sin⁡−1(sin⁡θ)=θ{\sin ^{ - 1}}\left( {\sin \theta } \right) = \theta . Therefore, we get
⇒cos⁡(sin⁡−1x)=cos⁡θ\Rightarrow \cos \left( {{{\sin }^{ - 1}}x} \right) = \cos \theta……………………(2)\left( 2 \right)
We know that sin⁡2θ+cos⁡2θ=1{\sin ^2}\theta + {\cos ^2}\theta = 1. Rewriting this identity, we get
cos⁡2θ=1−sin⁡2θ{\cos ^2}\theta = 1 - {\sin ^2}\theta
By taking the square root on both the sides, we get
⇒cos⁡θ=1−sin⁡2θ\Rightarrow \cos \theta = \sqrt {1 - {{\sin }^2}\theta }
By substituting equation (1)\left( 1 \right) in the above equation, we get
⇒cos⁡θ=1−x2\Rightarrow \cos \theta = \sqrt {1 - {x^2}}
Now substituting cos⁡θ=1−x2\cos \theta = \sqrt {1 - {x^2}} in equation (2)\left( 2 \right), we get
⇒cos⁡(sin⁡−1x)=1−x2\Rightarrow \cos \left( {{{\sin }^{ - 1}}x} \right) = \sqrt {1 - {x^2}}

Therefore, the solution of cos⁡(sin⁡−1x)\cos \left( {{{\sin }^{ - 1}}x} \right) is 1−x2\sqrt {1 - {x^2}} .

Note:
We know that Trigonometric Equation is defined as an equation involving trigonometric ratios. Trigonometric identity is an equation that is always true for all the variables. We should know that we have many trigonometric identities that are related to all the other trigonometric equations. Trigonometric Ratios of a Particular angle are the ratios of the sides of a right-angled triangle with respect to any of its acute angle. They are used to find the relationships between the sides of a right-angle triangle. Also, the value of the inverse trigonometric ratio should always lie in the domain of the trigonometric ratio.