`Lt_(x to (pi)/(2))(pi/2-x)tan x` is equal to (i) 1 (ii) `-1` (iii) `(pi)/(2)` (iv) `(2)/(pi)`

Options

A. 1
B. `-1`
C. `(pi)/(2)`
D. `(2)/(pi)`

Answer

To solve the limit \( \lim_{x \to \frac{\pi}{2}} \left( \frac{\pi}{2} - x \right) \tan x \), we can follow these steps:

### Step 1: Substitute \( y = \frac{\pi}{2} - x \)
Let \( y = \frac{\pi}{2} - x \). As \( x \) approaches \( \frac{\pi}{2} \), \( y \) approaches \( 0 \). Therefore, we can rewrite the limit in terms of \( y \):

\[ \lim_{x \to \frac{\pi}{2}} \left( \frac{\pi}{2} - x \right) \tan x = \lim_{y \to 0} y \tan\left(\frac{\pi}{2} - y\right) \]

### Step 2: Use the identity for tangent
Using the identity \( \tan\left(\frac{\pi}{2} - y\right) = \cot y \), we can rewrite the limit:

\[ \lim_{y \to 0} y \tan\left(\frac{\pi}{2} - y\right) = \lim_{y \to 0} y \cot y \]

### Step 3: Rewrite cotangent in terms of sine and cosine
Recall that \( \cot y = \frac{\cos y}{\sin y} \), so we have:

\[ \lim_{y \to 0} y \cot y = \lim_{y \to 0} y \cdot \frac{\cos y}{\sin y} = \lim_{y \to 0} \frac{y \cos y}{\sin y} \]

### Step 4: Evaluate the limit
As \( y \to 0 \), \( \cos y \to 1 \), so we can simplify:

\[ \lim_{y \to 0} \frac{y \cos y}{\sin y} = \lim_{y \to 0} \frac{y}{\sin y} \cdot \cos y \]

We know that \( \lim_{y \to 0} \frac{y}{\sin y} = 1 \). Therefore:

\[ \lim_{y \to 0} \frac{y \cos y}{\sin y} = 1 \cdot 1 = 1 \]

### Conclusion
Thus, we find that:

\[ \lim_{x \to \frac{\pi}{2}} \left( \frac{\pi}{2} - x \right) \tan x = 1 \]

The correct option is (i) 1.