Question

The slope of the tangent to the locus $y={\cos }^{-1}\left(\cos x\right)$ at $x=\frac{\pi }{4}$ is

• A. $1$
• B. $0$
• C. $2$
• D. $-1$

Answer 1

Answer: (A)

$1$

For $x=\frac{\pi }{4}y={\cos }^{-1}\cos \left(\frac{\pi }{4}\right)=\frac{\pi }{4}$

Therefore point is $(\frac{\pi }{4},\frac{\pi }{4})$

Now $y={\cos }^{-1}\cos \left(x\right)\Rightarrow \cos \left(y\right)=\cos \left(x\right)$

Slope is $\frac{dy}{dx}=\frac{-\sin \left(x\right)}{-\sin \left(y\right)}=1$