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The image contains a trigonometric equation that needs to be solved. The equation is:

\[
\frac{\sin x - \cos x}{\sin x + \cos x} + \frac{\sin x + \cos x}{\sin x - \cos x} = \frac{4\sin 2x}{\cos 2x}
\]

Let's go through the steps to solve this equation.

### Step 1: Simplify the Left-Hand Side (LHS)

The left-hand side (LHS) is:

\[
\frac{\sin x - \cos x}{\sin x + \cos x} + \frac{\sin x + \cos x}{\sin x - \cos x}
\]

To simplify this, notice that it is the sum of two fractions. Let's find a common denominator:

\[
\frac{(\sin x - \cos x)^2 + (\sin x + \cos x)^2}{(\sin x + \cos x)(\sin x - \cos x)}
\]

Now expand both squares:

\[
(\sin x - \cos x)^2 = \sin^2 x - 2\sin x\cos x + \cos^2 x
\]

\[
(\sin x + \cos x)^2 = \sin^2 x + 2\sin x\cos x + \cos^2 x
\]

Adding these, we get:

\[
(\sin^2 x - 2\sin x\cos x + \cos^2 x) + (\sin^2 x + 2\sin x\cos x + \cos^2 x) = 2\sin^2 x + 2\cos^2 x = 2(\sin^2 x + \cos^2 x)
\]

Since \(\sin^2 x + \cos^2 x = 1\), the numerator becomes \(2 \times 1 = 2\).

The denominator is the difference of squares:

\[
(\sin x + \cos x)(\sin x - \cos x) = \sin^2 x - \cos^2 x = \cos 2x
\]

So, the LHS simplifies to:

\[
\frac{2}{\cos 2x}
\]

### Step 2: Simplify the Right-Hand Side (RHS)

The right-hand side (RHS) is:

\[
\frac{4\sin 2x}{\cos 2x}
\]

### Step 3: Equate and Solve

We now equate the simplified LHS and RHS:

\[
\frac{2}{\cos 2x} = \frac{4\sin 2x}{\cos 2x}
\]

Since \(\cos 2x\) is common in both terms, and assuming \(\cos 2x \neq 0\):

\[
2 = 4\sin 2x
\]

Dividing both sides by 4:

\[
\sin 2x = \frac{1}{2}
\]

### Step 4: Find the Solutions

The general solution for \(\sin 2x = \frac{1}{2}\) is:

\[
2x = \frac{\pi}{6} + 2k\pi \quad \text{or} \quad 2x = \frac{5\pi}{6} + 2k\pi
\]

where \(k\) is any integer. Dividing by 2:

\[
x = \frac{\pi}{12} + k\pi \quad \text{or} \quad x = \frac{5\pi}{12} + k\pi
\]

These are the general solutions for \(x\).

Would you like more details or have any questions?

### Relative Questions:
1. What are the steps to solve trigonometric identities involving multiple angles?
2. How can we verify the solutions to a trigonometric equation?
3. What are the general solutions for the equation \(\sin x = \frac{1}{2}\)?
4. How does the identity \(\sin^2 x + \cos^2 x = 1\) simplify trigonometric equations?
5. What is the significance of the double-angle formula in solving trigonometric equations?

### Tip:
Always check for extraneous solutions when solving trigonometric equations, especially when dividing by trigonometric functions.

Simplify and solve the trigonometric equation: (sin(x) - cos(x))/(sin(x) + cos(x)) + (sin(x) + cos(x))/(sin(x) - cos(x)) = 4sin(2x)/cos(2x)

Learn how to simplify and solve a trigonometric equation involving multiple identities, including the Pythagorean Identity and Double-Angle Formulas. This step-by-step solution is suitable for students in Grades 10-12.

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