If `theta in((pi)/(4),(pi)/(2)) and f(theta)=sec 2theta- tan 2theta`, then `f((pi)/(4)-theta)=`

Options

A. `tan theta`
B. `cottheta`
C. `sec 2 theta`
D. `tan2theta`

Answer

To solve the problem, we need to find the value of \( f\left(\frac{\pi}{4} - \theta\right) \) given that \( f(\theta) = \sec(2\theta) - \tan(2\theta) \).

### Step-by-Step Solution:

1. **Identify the Function**:
We start with the function:
\[ f(\theta) = \sec(2\theta) - \tan(2\theta) \]

2. **Substitute \( \frac{\pi}{4} - \theta \)**:
We need to evaluate:
\[ f\left(\frac{\pi}{4} - \theta\right) = \sec\left(2\left(\frac{\pi}{4} - \theta\right)\right) - \tan\left(2\left(\frac{\pi}{4} - \theta\right)\right) \]

3. **Simplify the Angles**:
Calculate \( 2\left(\frac{\pi}{4} - \theta\right) \):
\[ 2\left(\frac{\pi}{4} - \theta\right) = \frac{\pi}{2} - 2\theta \]
Thus, we rewrite \( f\left(\frac{\pi}{4} - \theta\right) \):
\[ f\left(\frac{\pi}{4} - \theta\right) = \sec\left(\frac{\pi}{2} - 2\theta\right) - \tan\left(\frac{\pi}{2} - 2\theta\right) \]

4. **Use Trigonometric Identities**:
Using the identities:
\[ \sec\left(\frac{\pi}{2} - x\right) = \csc(x) \quad \text{and} \quad \tan\left(\frac{\pi}{2} - x\right) = \cot(x) \]
We can rewrite:
\[ f\left(\frac{\pi}{4} - \theta\right) = \csc(2\theta) - \cot(2\theta) \]

5. **Express in Terms of Sine and Cosine**:
Recall that:
\[ \csc(2\theta) = \frac{1}{\sin(2\theta)} \quad \text{and} \quad \cot(2\theta) = \frac{\cos(2\theta)}{\sin(2\theta)} \]
Thus:
\[ f\left(\frac{\pi}{4} - \theta\right) = \frac{1 - \cos(2\theta)}{\sin(2\theta)} \]

6. **Use the Identity for Sine**:
We know that:
\[ 1 - \cos(2\theta) = 2\sin^2(\theta) \]
Therefore:
\[ f\left(\frac{\pi}{4} - \theta\right) = \frac{2\sin^2(\theta)}{\sin(2\theta)} \]

7. **Substitute for \( \sin(2\theta) \)**:
Using the double angle identity:
\[ \sin(2\theta) = 2\sin(\theta)\cos(\theta) \]
We can simplify:
\[ f\left(\frac{\pi}{4} - \theta\right) = \frac{2\sin^2(\theta)}{2\sin(\theta)\cos(\theta)} = \frac{\sin(\theta)}{\cos(\theta)} = \tan(\theta) \]

### Final Result:
Thus, we find that:
\[ f\left(\frac{\pi}{4} - \theta\right) = \tan(\theta) \]