Write down the value of ` "cosec"^(-1)x +sec^(-1) x , ` when ` x ge 1 ` .

Answer

To find the value of \( \csc^{-1} x + \sec^{-1} x \) when \( x \geq 1 \), we can follow these steps:

### Step 1: Understand the definitions
The inverse cosecant function \( \csc^{-1} x \) is defined as the angle \( \theta \) such that \( \csc \theta = x \). Similarly, the inverse secant function \( \sec^{-1} x \) is defined as the angle \( \phi \) such that \( \sec \phi = x \).

### Step 2: Relate secant to cosine
Since \( \sec \phi = \frac{1}{\cos \phi} \), we can express \( \sec^{-1} x \) in terms of cosine:
\[ \sec^{-1} x = \cos^{-1} \left( \frac{1}{x} \right) \]

### Step 3: Relate cosecant to sine
For \( \csc^{-1} x \), we have:
\[ \csc^{-1} x = \sin^{-1} \left( \frac{1}{x} \right) \]

### Step 4: Combine the two inverse functions
Now we can write:
\[ \csc^{-1} x + \sec^{-1} x = \sin^{-1} \left( \frac{1}{x} \right) + \cos^{-1} \left( \frac{1}{x} \right) \]

### Step 5: Use the identity
We know from trigonometric identities that:
\[ \sin^{-1} y + \cos^{-1} y = \frac{\pi}{2} \quad \text{for } 0 < y < 1 \]
In our case, \( y = \frac{1}{x} \), and since \( x \geq 1 \), we have \( 0 < \frac{1}{x} \leq 1 \).

### Step 6: Conclude the result
Thus, we can conclude:
\[ \csc^{-1} x + \sec^{-1} x = \frac{\pi}{2} \]

### Final Answer:
\[ \csc^{-1} x + \sec^{-1} x = \frac{\pi}{2} \quad \text{for } x \geq 1 \]

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