To solve the equation \( \tan \theta = \sin 2\theta \), we will start by rewriting the trigonometric functions in terms of sine and cosine.

### Step 1: Rewrite \( \tan \theta \) and \( \sin 2\theta \)
\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\]
\[
\sin 2\theta = 2 \sin \theta \cos \theta
\]

So, the equation becomes:
\[
\frac{\sin \theta}{\cos \theta} = 2 \sin \theta \cos \theta
\]

### Step 2: Simplify the equation
Assume \( \sin \theta \neq 0 \) (we will consider \( \sin \theta = 0 \) later). Divide both sides by \( \sin \theta \):
\[
\frac{1}{\cos \theta} = 2 \cos \theta
\]

Multiply both sides by \( \cos \theta \) to get:
\[
1 = 2 \cos^2 \theta
\]

### Step 3: Solve for \( \cos^2 \theta \)
\[
\cos^2 \theta = \frac{1}{2}
\]

Take the square root of both sides:
\[
\cos \theta = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}
\]

### Step 4: Find \( \theta \)
The general solutions for \( \cos \theta = \pm \frac{\sqrt{2}}{2} \) are:
\[
\theta = \frac{\pi}{4} + 2n\pi \quad \text{or} \quad \theta = \frac{3\pi}{4} + 2n\pi \quad \text{for integer } n
\]

### Step 5: Consider \( \sin \theta = 0 \)
If \( \sin \theta = 0 \), then \( \theta = n\pi \), where \( n \) is an integer. However, in this case, \( \sin 2\theta = 0 \) as well, which means the original equation \( \tan \theta = \sin 2\theta \) is satisfied.

### Final Answer:
The solutions to the equation are:
\[
\theta = n\pi \quad \text{or} \quad \theta = \frac{\pi}{4} + 2n\pi \quad \text{or} \quad \theta = \frac{3\pi}{4} + 2n\pi \quad \text{where } n \text{ is an integer}.
\]

Would you like more details on any step or have any questions?

Here are five related questions:
1. How do you derive the double-angle identity for sine?
2. What is the geometric interpretation of \( \tan \theta \) and \( \sin 2\theta \)?
3. How do we derive the general solution for trigonometric equations?
4. Can you solve similar equations involving \( \cos \theta \) and \( \sin \theta \)?
5. What are the applications of solving such trigonometric equations?

**Tip:** Always check for special cases where the trigonometric functions may be undefined, such as when \( \cos \theta = 0 \) for \( \tan \theta \).

Discover how to solve the trigonometric equation tanθ = sin2θ using identities such as tanθ = sinθ / cosθ and the double-angle identity for sine. This detailed step-by-step solution is suitable for high school students in Grades 10-12 and covers key trigonometric concepts and formulas.

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