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Question: The value of \[{{\cos }^{-1}}(-1)-{{\sin }^{-1}}(1)\] is: A. \[\pi \] B. \[\dfrac{\pi }{2}\] C...

The value of cos⁡−1(−1)−sin⁡−1(1){{\cos }^{-1}}(-1)-{{\sin }^{-1}}(1) is:
A. π\pi
B. π2\dfrac{\pi }{2}
C. 3π2\dfrac{3\pi }{2}
D. −3π2-\dfrac{3\pi }{2}

Explanation

Solution

Hint: For the above question we will have to know about cos⁡−1x{{\cos }^{-1}}x and sin−1xsi{{n}^{-1}}x are the inverse trigonometric function. The domain of both cos⁡−1x{{\cos }^{-1}}x and sin−1xsi{{n}^{-1}}x is [−1,1]\left[ -1,1 \right] and the range of sin−1xsi{{n}^{-1}}x is [−π2,π2]\left[ \dfrac{-\pi }{2},\dfrac{\pi }{2} \right] and the range of cos⁡−1x{{\cos }^{-1}}x is [0,π]\left[ 0,\pi \right]. So we will substitute the value of cos⁡−1(−1){{\cos }^{-1}}(-1) and sin⁡−1(1){{\sin }^{-1}}(1) in the given expression to get the answer.

Complete step-by-step answer:
We have been given cos⁡−1(−1)−sin⁡−1(1){{\cos }^{-1}}(-1)-{{\sin }^{-1}}(1).
We know that the range of cos⁡−1x{{\cos }^{-1}}x is [0,π]\left[ 0,\pi \right].
We also know that cosy=cos⁡(cos⁡−1x)=xcosy=\cos \left( {{\cos }^{-1}}x \right)=x.
Let y=cos⁡−1(−1)y={{\cos }^{-1}}(-1).
On taking cosine both the sides, we get as follows:

& \cos y=\cos \left[ {{\cos }^{-1}}(-1) \right] \\\ & \cos y=-1 \\\ \end{aligned}$$ $$\Rightarrow y=\pi $$ since $$\cos \pi =-1$$. Also, the value of $$\pi $$ is in the range of $${{\cos }^{-1}}x$$. Once again, the range of $$si{{n}^{-1}}x$$ is $$\left[ \dfrac{-\pi }{2},\dfrac{\pi }{2} \right]$$. Also, we know that $$\sin y=\sin \left( {{\sin }^{-1}}x \right)=x$$. Let $$y={{\sin }^{-1}}1$$. On taking sin both the sides, we get as follows: $$\begin{aligned} & \sin y=\sin \left( {{\sin }^{-1}}1 \right) \\\ & \Rightarrow \sin y=1 \\\ \end{aligned}$$ $$\Rightarrow y=\dfrac{\pi }{2}$$ since $$\sin \dfrac{\pi }{2}=1$$. Also, the value of $$\dfrac{\pi }{2}$$ is in the range of $${{\sin }^{-1}}x$$. Now substituting these values in the expression, we get as follows: $${{\cos }^{-1}}(-1)-{{\sin }^{-1}}(1)=\pi -\dfrac{\pi }{2}=\dfrac{2\pi -\pi }{2}=\dfrac{\pi }{2}$$ Therefore the correct answer is option B. Note: Just check the value of the inverse function that it must lie in the range of that inverse function otherwise we use the transformation to get the value. Don’t make a mistake like $${{\cos }^{-1}}(-1)=-\pi $$ as it is a chance of doing it if you are in a hurry.