Question

Assertion (A) ${\cos }^{-1}x\ge {\sin }^{-1}x,\forall x\in [-1,1]$
Reason (R) ${\cos }^{-1}x$ is decreasing function $\forall x\in [-1,1]$

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true and R is not correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The correct option is D A is false but R is true

As we can see from the graph, $co{s}^{-1}\left(x\right)$ is a decreasing function while $si{n}^{-1}\left(x\right)$ is an increasing function on $[-1,1]$ and $si{n}^{-1}\left(x\right)=co{s}^{-1}\left(x\right)$ at $x=\frac{1}{\sqrt{2}}$
Hence $co{s}^{-1}x\ge si{n}^{-1}\left(x\right)$ on $[-1,1]$ is false.