`lim_(theta to pi//2) (sec theta - tan theta) =`

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To solve the limit \( \lim_{\theta \to \frac{\pi}{2}} (\sec \theta - \tan \theta) \), we will follow these steps:

### Step 1: Substitute the limit
We begin by substituting \( \theta = \frac{\pi}{2} \) into the expression:
\[ \sec\left(\frac{\pi}{2}\right) - \tan\left(\frac{\pi}{2}\right) \]
Calculating these values, we find:
\[ \sec\left(\frac{\pi}{2}\right) = \frac{1}{\cos\left(\frac{\pi}{2}\right)} = \frac{1}{0} \quad (\text{undefined, approaches } \infty) \]
\[ \tan\left(\frac{\pi}{2}\right) = \frac{\sin\left(\frac{\pi}{2}\right)}{\cos\left(\frac{\pi}{2}\right)} = \frac{1}{0} \quad (\text{undefined, approaches } \infty) \]
Thus, we have:
\[ \sec\left(\frac{\pi}{2}\right) - \tan\left(\frac{\pi}{2}\right) = \infty - \infty \quad (\text{indeterminate form}) \]

### Step 2: Rewrite the expression
To resolve the indeterminate form, we can rewrite the expression:
\[ \sec \theta - \tan \theta = \frac{1 - \sin \theta}{\cos \theta} \]
This can be rewritten as:
\[ \frac{\sec^2 \theta - \tan^2 \theta}{\sec \theta + \tan \theta} \]
Using the identity \( \sec^2 \theta - \tan^2 \theta = 1 \):
\[ \sec \theta - \tan \theta = \frac{1}{\sec \theta + \tan \theta} \]

### Step 3: Substitute again
Now, we substitute \( \theta \to \frac{\pi}{2} \):
\[ \sec\left(\frac{\pi}{2}\right) + \tan\left(\frac{\pi}{2}\right) = \infty + \infty = \infty \]
Thus, we have:
\[ \lim_{\theta \to \frac{\pi}{2}} (\sec \theta - \tan \theta) = \lim_{\theta \to \frac{\pi}{2}} \frac{1}{\sec \theta + \tan \theta} = \frac{1}{\infty} = 0 \]

### Conclusion
Therefore, the limit is:
\[ \lim_{\theta \to \frac{\pi}{2}} (\sec \theta - \tan \theta) = 0 \]