Question

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

Answer 1

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

Complete step-by-step answer:
We have been given \[{{\cos }^{-1}}(-1)-{{\sin }^{-1}}(1)\].
We know that the range of \[{{\cos }^{-1}}x\] is \[\left[ 0,\pi \right]\].
We also know that \[cosy=\cos \left( {{\cos }^{-1}}x \right)=x\].
Let \[y={{\cos }^{-1}}(-1)\].
On taking cosine both the sides, we get as follows:
\[\begin{align}
  & \cos y=\cos \left[ {{\cos }^{-1}}(-1) \right] \\
 & \cos y=-1 \\
\end{align}\]
\[\Rightarrow y=\pi \] 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:
\[\begin{align}
  & \sin y=\sin \left( {{\sin }^{-1}}1 \right) \\
 & \Rightarrow \sin y=1 \\
\end{align}\]
\[\Rightarrow y=\dfrac{\pi }{2}\] 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.