Evaluate the following (10-17) limits :
`lim_(x to pi//2)(cos^(2)x)/(1-sin^(2)x)`.

Answer

To evaluate the limit

\[ \lim_{x \to \frac{\pi}{2}} \frac{\cos^2 x}{1 - \sin^2 x}, \]

we can follow these steps:

### Step 1: Simplify the expression
We know from the Pythagorean identity that

\[ \sin^2 x + \cos^2 x = 1. \]

This implies that

\[ 1 - \sin^2 x = \cos^2 x. \]

### Step 2: Substitute the identity into the limit
Now, we can substitute \(1 - \sin^2 x\) with \(\cos^2 x\):

\[ \lim_{x \to \frac{\pi}{2}} \frac{\cos^2 x}{1 - \sin^2 x} = \lim_{x \to \frac{\pi}{2}} \frac{\cos^2 x}{\cos^2 x}. \]

### Step 3: Cancel the common terms
Since \(\cos^2 x\) is in both the numerator and the denominator, we can cancel them out (as long as \(\cos^2 x \neq 0\)):

\[ \lim_{x \to \frac{\pi}{2}} \frac{\cos^2 x}{\cos^2 x} = \lim_{x \to \frac{\pi}{2}} 1. \]

### Step 4: Evaluate the limit
The limit of a constant is simply the constant itself:

\[ \lim_{x \to \frac{\pi}{2}} 1 = 1. \]

### Final Answer
Thus, the value of the limit is

\[ \boxed{1}. \]