Evaluate `tan^(-1) 1 +cos^(-1) . 1/3 +sin^(-1). 1/3 `

Answer

To evaluate the expression \( \tan^{-1}(1) + \cos^{-1}\left(\frac{1}{3}\right) + \sin^{-1}\left(\frac{1}{3}\right) \), we can follow these steps:

### Step 1: Evaluate \( \tan^{-1}(1) \)
The value of \( \tan^{-1}(1) \) is known to be \( \frac{\pi}{4} \) because the tangent of \( \frac{\pi}{4} \) is 1.

### Step 2: Use the identity for \( \cos^{-1}(x) + \sin^{-1}(x) \)
We can use the identity:
\[ \cos^{-1}(x) + \sin^{-1}(x) = \frac{\pi}{2} \]
for \( x = \frac{1}{3} \). Thus,
\[ \cos^{-1}\left(\frac{1}{3}\right) + \sin^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{2} \]

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

### Step 4: Find a common denominator and add
To add \( \frac{\pi}{4} \) and \( \frac{\pi}{2} \), we need a common denominator. The least common multiple of the denominators (4 and 2) is 4. Therefore, we convert \( \frac{\pi}{2} \) to have a denominator of 4:
\[ \frac{\pi}{2} = \frac{2\pi}{4} \]
Now we can add:
\[ \frac{\pi}{4} + \frac{2\pi}{4} = \frac{3\pi}{4} \]

### Final Answer
Thus, the final answer is:
\[ \frac{3\pi}{4} \]

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