Find the value of : `cosec[sec^(-1)(sqrt(2))+cot^(-1)(1)]`

Explanation

To solve the expression \( \csc\left[\sec^{-1}(\sqrt{2}) + \cot^{-1}(1)\right] \), we will follow these steps:

### Step 1: Evaluate \( \sec^{-1}(\sqrt{2}) \)

The value \( \sec^{-1}(\sqrt{2}) \) corresponds to the angle whose secant is \( \sqrt{2} \). We know that:

\[ \sec\left(\frac{\pi}{4}\right) = \sqrt{2} \]

Thus,

\[ \sec^{-1}(\sqrt{2}) = \frac{\pi}{4} \]

### Step 2: Evaluate \( \cot^{-1}(1) \)

The value \( \cot^{-1}(1) \) corresponds to the angle whose cotangent is \( 1 \). We know that:

\[ \cot\left(\frac{\pi}{4}\right) = 1 \]

Thus,

\[ \cot^{-1}(1) = \frac{\pi}{4} \]

### Step 3: Add the angles

Now we can add the two angles we found:

\[ \sec^{-1}(\sqrt{2}) + \cot^{-1}(1) = \frac{\pi}{4} + \frac{\pi}{4} = \frac{2\pi}{4} = \frac{\pi}{2} \]

### Step 4: Evaluate \( \csc\left(\frac{\pi}{2}\right) \)

Now we need to find \( \csc\left(\frac{\pi}{2}\right) \). We know that:

\[ \sin\left(\frac{\pi}{2}\right) = 1 \]

Thus,

\[ \csc\left(\frac{\pi}{2}\right) = \frac{1}{\sin\left(\frac{\pi}{2}\right)} = \frac{1}{1} = 1 \]

### Final Answer

Therefore, the value of \( \csc\left[\sec^{-1}(\sqrt{2}) + \cot^{-1}(1)\right] \) is:

\[ \boxed{1} \]