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

Answer

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

### Step 1: Understand the Inverse Functions
We know that:
- \( \sec^{-1}(x) \) gives us the angle whose secant is \( x \).
- \( \csc^{-1}(y) \) gives us the angle whose cosecant is \( y \).

### Step 2: Find \( \sec^{-1}(2) \)
The value of \( \sec^{-1}(2) \) corresponds to the angle \( \theta \) such that \( \sec(\theta) = 2 \). This can be rewritten in terms of cosine:
\[ \sec(\theta) = \frac{1}{\cos(\theta)} \implies \cos(\theta) = \frac{1}{2} \]
The angle \( \theta \) that satisfies this is:
\[ \theta = \frac{\pi}{3} \]
Thus, we have:
\[ \sec^{-1}(2) = \frac{\pi}{3} \]

### Step 3: Substitute Back into the Equation
Now we substitute \( \sec^{-1}(2) \) back into the original equation:
\[ \frac{\pi}{3} + \csc^{-1}(y) = \frac{\pi}{2} \]

### Step 4: Isolate \( \csc^{-1}(y) \)
To isolate \( \csc^{-1}(y) \), we subtract \( \frac{\pi}{3} \) from both sides:
\[ \csc^{-1}(y) = \frac{\pi}{2} - \frac{\pi}{3} \]

### Step 5: Simplify the Right Side
To simplify \( \frac{\pi}{2} - \frac{\pi}{3} \), we need a common denominator, which is 6:
\[ \frac{\pi}{2} = \frac{3\pi}{6}, \quad \frac{\pi}{3} = \frac{2\pi}{6} \]
So,
\[ \csc^{-1}(y) = \frac{3\pi}{6} - \frac{2\pi}{6} = \frac{\pi}{6} \]

### Step 6: Find \( y \)
Now we know that:
\[ \csc^{-1}(y) = \frac{\pi}{6} \]
This means:
\[ \csc\left(\frac{\pi}{6}\right) = y \]
From trigonometric values, we know:
\[ \csc\left(\frac{\pi}{6}\right) = \frac{1}{\sin\left(\frac{\pi}{6}\right)} = \frac{1}{\frac{1}{2}} = 2 \]
Thus, we find:
\[ y = 2 \]

### Final Answer
The value of \( y \) is \( 2 \).

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