Find the value of
`sin^(-1)("sin"(2pi)/(3))+cos^(-1)("cos"(4pi)/(3))`

Answer

To find the value of \( \sin^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right) + \cos^{-1}\left(\cos\left(\frac{4\pi}{3}\right)\right) \), we can follow these steps:

### Step 1: Simplify \( \sin^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right) \)

The value of \( \frac{2\pi}{3} \) is in the second quadrant, where the sine function is positive. The sine inverse function returns values in the range \( \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \). To find the equivalent angle in this range, we can use the identity:
\[ \sin^{-1}(\sin(\theta)) = \theta \quad \text{if } \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \]
However, since \( \frac{2\pi}{3} \) is not in this range, we can find its reference angle:
\[ \sin^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right) = \pi - \frac{2\pi}{3} = \frac{\pi}{3} \]

### Step 2: Simplify \( \cos^{-1}\left(\cos\left(\frac{4\pi}{3}\right)\right) \)

The angle \( \frac{4\pi}{3} \) is in the third quadrant, where the cosine function is negative. The cosine inverse function returns values in the range \( [0, \pi] \). To find the equivalent angle in this range, we can use the identity:
\[ \cos^{-1}(\cos(\theta)) = \theta \quad \text{if } \theta \in [0, \pi] \]
Since \( \frac{4\pi}{3} \) is not in this range, we can find its reference angle:
\[ \cos^{-1}\left(\cos\left(\frac{4\pi}{3}\right)\right) = 2\pi - \frac{4\pi}{3} = \frac{2\pi}{3} \]

### Step 3: Combine the results

Now we can add the results from Step 1 and Step 2:
\[ \sin^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right) + \cos^{-1}\left(\cos\left(\frac{4\pi}{3}\right)\right) = \frac{\pi}{3} + \frac{2\pi}{3} = \pi \]

### Final Answer
Thus, the value of \( \sin^{-1}\left(\sin\left(\frac{2\pi}{3}\right)\right) + \cos^{-1}\left(\cos\left(\frac{4\pi}{3}\right)\right) \) is:
\[ \boxed{\pi} \]