Question

The value of \(\cos(\sin^{-1}\frac{\pi}{3}+\cos^{-1}\frac{\pi}{3})\) is

• A. 0
• B. 1
• C. -1
• D. Does not exist

Answer 1

The Correct Option is D

We need to find the value of \(\cos(\sin^{-1}(\frac{\pi}{3}) + \cos^{-1}(\frac{\pi}{3}))\).

First, note that the domain of both \(\sin^{-1}(x)\) and \(\cos^{-1}(x)\) is \([-1, 1]\). However, \(\frac{\pi}{3} \approx \frac{3.14}{3} > 1\). Therefore, \(\sin^{-1}(\frac{\pi}{3})\) and \(\cos^{-1}(\frac{\pi}{3})\) are not defined.

Since \(\sin^{-1}(\frac{\pi}{3})\) and \(\cos^{-1}(\frac{\pi}{3})\) do not exist, the entire expression does not exist.

Thus, the correct option is (D) Does not exist.

Answer 2

\[ \cos\left( \sin^{-1}\left( \frac{\pi}{3} \right) + \cos^{-1}\left( \frac{\pi}{3} \right) \right) \] 
 - \( \sin^{-1}\left( \frac{\pi}{3} \right) \) and \( \cos^{-1}\left( \frac{\pi}{3} \right) \) are angles whose sine and cosine are \( \frac{\pi}{3} \), respectively. 

However, the values of \( \sin^{-1}(x) \) and \( \cos^{-1}(x) \) are defined only for arguments between \(-1\) and \( 1\). 

Since \( \frac{\pi}{3} \approx 1.047 \) is greater than 1, neither \( \sin^{-1} \left( \frac{\pi}{3} \right) \) nor \( \cos^{-1} \left( \frac{\pi}{3} \right) \) is defined.  

The value does not exist.