Question

$\cos \left[\cos ^{-1}(-1 / 7)+\sin ^{-1}(-1 / 7)\right]=$
$1)$-1 / 3
2) 0
3)$1 / 3$
4)$4 / 9$

Answer 1

Hint: First we need to compare the given expression with the general expression to find the value in the place of x. After that, using trigonometric identities we can easily find the solution to the given expression.

Formula Used:
The general equation is $\sin ^{-1}(x)+\cos ^{-1}(x)=\pi / 2$

Complete step by step Solution:
Given that
$\cos \left[\cos ^{-1}(-1 / 7)+\sin ^{-1}(-1 / 7)\right]$
We know that the general equation
$\sin ^{-1}(x)+\cos ^{-1}(x)=\pi / 2$
Here in the place of $\sin ^{-1}(x)+\cos ^{-1}(x)=\pi / 2$ we have $(-1/7)$
So accordingly we can say that the resultant value of the given equation is $\cos \left[\cos ^{-1}(-1 / 7)+\sin ^{-1}(-1 / 7)\right]=\cos \pi / 2$
Thus,
 $\cos \left[\cos ^{-1}(-1 / 7)+\sin ^{-1}(-1 / 7)\right]=\cos \pi / 2$
 $=0$
Finding the coordinates of the equivalent point (0, 1) on the unit circle and creating an angle of $\pi /2$ radians with the x-axis will yield the value of $\cos \left[\cos ^{-1}(-1 / 7)+\sin ^{-1}(-1 / 7)\right]=\cos \pi / 2$. The x-coordinate is equivalent to the value of 0. ∴ $\cos \pi /2=0$.

Hence, the correct option is 2.

Additional Information: The domain and range of the functions determine the characteristics of inverse trigonometric functions. A greater comprehension of this idea and the ability to solve difficulties both depend on certain aspects of inverse trigonometric functions. Recall that "Arc Functions" is another name for inverse trigonometric functions. They generate the length of the arc required to arrive at a specific value for a given trigonometric function. The range of values that an inverse function is capable of with its specified domain is referred to as the inverse function's range. The collection of all potential independent variables where a function exists is referred to as the function's domain. There is a specific range in which inverse trigonometric functions are defined.

Note: Students should keep in mind all the formulas related to inverse trigonometric functions for solving such types of questions.