Let `0lt x lepi//4, (sec 2x-tan2x)` equals

Options

A. `tan^(2)(x+pi//4)`
B. `tan(x+pi//4)`
C. `tan(pi//4-x)`
D. `tan(x-pi//4)`

Answer

To solve the equation \( \sec 2x - \tan 2x = 2 \) for \( 0 < x \leq \frac{\pi}{4} \), we will follow these steps:

### Step 1: Rewrite the trigonometric functions
We start by rewriting \( \sec 2x \) and \( \tan 2x \) in terms of sine and cosine:
\[ \sec 2x = \frac{1}{\cos 2x}, \quad \tan 2x = \frac{\sin 2x}{\cos 2x} \]
Thus, we can rewrite the equation as:
\[ \frac{1}{\cos 2x} - \frac{\sin 2x}{\cos 2x} = 2 \]

### Step 2: Combine the fractions
Now, we can combine the fractions over a common denominator:
\[ \frac{1 - \sin 2x}{\cos 2x} = 2 \]

### Step 3: Cross-multiply
Next, we cross-multiply to eliminate the fraction:
\[ 1 - \sin 2x = 2 \cos 2x \]

### Step 4: Rearrange the equation
Rearranging gives us:
\[ 1 = 2 \cos 2x + \sin 2x \]

### Step 5: Use the Pythagorean identity
We can use the Pythagorean identity \( \cos^2 2x + \sin^2 2x = 1 \). However, we will first express \( \sin 2x \) in terms of \( \cos 2x \):
Let \( \sin 2x = \sqrt{1 - \cos^2 2x} \).

### Step 6: Substitute and simplify
Substituting \( \sin 2x \) into the equation gives:
\[ 1 = 2 \cos 2x + \sqrt{1 - \cos^2 2x} \]
Squaring both sides to eliminate the square root leads to:
\[ (1 - 2 \cos 2x)^2 = 1 - \cos^2 2x \]

### Step 7: Expand and simplify
Expanding the left side:
\[ 1 - 4 \cos 2x + 4 \cos^2 2x = 1 - \cos^2 2x \]
This simplifies to:
\[ 4 \cos^2 2x - 4 \cos 2x = 0 \]

### Step 8: Factor the equation
Factoring gives:
\[ 4 \cos 2x (\cos 2x - 1) = 0 \]
This results in two cases:
1. \( \cos 2x = 0 \)
2. \( \cos 2x - 1 = 0 \) (i.e., \( \cos 2x = 1 \))

### Step 9: Solve for \( x \)
1. For \( \cos 2x = 0 \):
\[ 2x = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \implies x = \frac{\pi}{4} + \frac{n\pi}{2} \]
Since \( 0 < x \leq \frac{\pi}{4} \), we take \( n = 0 \) which gives \( x = \frac{\pi}{4} \).

2. For \( \cos 2x = 1 \):
\[ 2x = 2n\pi \quad (n \in \mathbb{Z}) \implies x = n\pi \]
The only solution in the interval \( 0 < x \leq \frac{\pi}{4} \) is \( x = 0 \), which is not included in our range.

### Conclusion
Thus, the only solution is:
\[ x = \frac{\pi}{4} \]