https://r2.erweima.ai/mathbot/image/1719934229424yymijb6h.JPG

Let's continue solving the questions from the image provided.

### Question (iii):
\[ \sin 2A = 2 \sin A \text{ is true when } A \text{ is:} \]

We already solved this in the previous explanation:
For \(\sin 2A = 2 \sin A\) to hold true, \(\cos A = 1\), which happens when:
\[ A = 0^\circ \]

So, the correct option is:
\[ \boxed{\text{(A) 0}^\circ} \]

### Question (iv):
\[ \frac{2 \tan 30^\circ}{1 - \tan^2 30^\circ} \]

We already solved this in the previous explanation:
\[ \frac{2 \cdot \frac{1}{\sqrt{3}}}{1 - \left(\frac{1}{\sqrt{3}}\right)^2} = \sqrt{3} \]

So, the correct option is:
\[ \boxed{\text{(C) tan } 60^\circ} \]

### Question 3:
\[ \text{If } \tan (A + B) = \sqrt{3} \text{ and } \tan (A - B) = \frac{1}{\sqrt{3}}, \text{ find the values of } A \text{ and } B. \]

1. **Given:**
   \[ \tan (A + B) = \sqrt{3} \implies A + B = 60^\circ \]
   \[ \tan (A - B) = \frac{1}{\sqrt{3}} \implies A - B = 30^\circ \]

2. **Solving the system of equations:**
   \[
   \begin{cases}
   A + B = 60^\circ \\
   A - B = 30^\circ
   \end{cases}
   \]

3. **Add the equations:**
   \[ 2A = 90^\circ \implies A = 45^\circ \]

4. **Subtract the equations:**
   \[ 2B = 30^\circ \implies B = 15^\circ \]

So, the values of \( A \) and \( B \) are:
\[ \boxed{A = 45^\circ, B = 15^\circ} \]

### Question 4:
**Identify which statements are true or false with reasons:**

(i) \( \sin (A + B) = \sin A + \sin B \)

**False**: The sine of a sum is not equal to the sum of sines. The correct formula is:
\[ \sin (A + B) = \sin A \cos B + \cos A \sin B \]

(ii) \( \theta \) increases, the value of \(\sin \theta\) also increases.

**True** for \(0^\circ \leq \theta \leq 90^\circ\), as \(\sin \theta\) increases from 0 to 1.

(iii) \( \theta \) increases, the value of \(\cos \theta\) also increases.

**False**: For \(0^\circ \leq \theta \leq 90^\circ\), \(\cos \theta\) decreases from 1 to 0.

(iv) \(\theta\) increases, the value of \(\cos \theta\) also increases.

**False**: As mentioned, \(\cos \theta\) decreases as \(\theta\) increases from 0 to 90 degrees.

(v) \( \theta \) increases, the value of \(\sin \theta = \cos \theta\).

**True** at \(\theta = 45^\circ\), but this is a specific case.

(vi) At \( \theta = 0^\circ \), \(\cot \theta\) is undefined.

**True**: \(\cot \theta = \frac{1}{\tan \theta} \) and \(\tan 0^\circ = 0\), making \(\cot 0^\circ\) undefined.

Would you like more details or have any questions?

Here are 5 questions you can ask:
1. How do you derive the double angle formula for sine?
2. What are some common trigonometric identities?
3. Can you explain the steps to solve another trigonometric expression?
4. What are the values of sine and cosine for standard angles?
5. How can we use trigonometric identities to simplify expressions?

**Tip:** When solving trigonometric equations, always consider the ranges of angles and the periodicity of trigonometric functions.

Explore solutions to trigonometric equations and identities including double angle formulas for sine and sum/difference identities for tangent. Learn key theorems and formulas essential for high school trigonometry.

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