If `sec^(-1)(x) +"cosec"^(-1) (1/3) = pi/2 ` , then find x .

Answer

To solve the equation \( \sec^{-1}(x) + \csc^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{2} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sec^{-1}(x) + \csc^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{2} \]
We can isolate \( \sec^{-1}(x) \):
\[ \sec^{-1}(x) = \frac{\pi}{2} - \csc^{-1}\left(\frac{1}{3}\right) \]

### Step 2: Use the complementary angle identity
Using the identity \( \sec\left(\frac{\pi}{2} - \theta\right) = \csc(\theta) \), we can rewrite the right-hand side:
\[ \sec^{-1}(x) = \csc\left(\csc^{-1}\left(\frac{1}{3}\right)\right) \]

### Step 3: Evaluate \( \csc\left(\csc^{-1}\left(\frac{1}{3}\right)\right) \)
The value of \( \csc\left(\csc^{-1}\left(\frac{1}{3}\right)\right) \) is simply \( \frac{1}{3} \) because the cosecant function and its inverse cancel each other out:
\[ \sec^{-1}(x) = \frac{1}{3} \]

### Step 4: Convert to secant form
Now, we can convert this back to the secant function:
\[ x = \sec\left(\frac{1}{3}\right) \]

### Step 5: Find the value of \( x \)
To find \( x \), we need to compute \( \sec\left(\frac{1}{3}\right) \):
\[ x = \frac{1}{\cos\left(\frac{1}{3}\right)} \]

### Conclusion
Thus, the value of \( x \) is:
\[ x = \sec\left(\csc^{-1}\left(\frac{1}{3}\right)\right) = \frac{1}{\cos\left(\frac{1}{3}\right)} \]