1. If \( \alpha \) and \( \beta \) are complementary angles, then prove that \[ \cot \beta + \cos \beta = \frac{\cos \beta}{\cos \alpha (1 + \sin \beta)} \]

2. If \(A\) and \(B\) are complementary angles, prove that: \[ \frac{\sin A + \sin B}{\cos B - \cos A} = \frac{\cos A + \cos B}{\sin A - \sin B} \]

3. Given: \(\tan \theta + \sin \theta = m\) and \(\tan \theta - \sin \theta = n\) Prove that: \[ m^2 - n^2 = 4\sqrt{mn} \quad \text{where } 0^\circ < \theta < 90^\circ \]

4. If \(A\) and \(B\) are two complementary angles, then show that \[ (\sin A + \sin B)^2 = 1 + 2\sin A \sin B \]

5. If a cone has volume \(V\) cubic units, total surface area \(S\) square units, height \(h\) units, and base radius \(r\) units, then show that: \[ S = 2V\left(\frac{1}{h} + \frac{1}{r}\right) \]

6. In triangle △ABC, ∠A is a right angle and BP and CQ are medians. Prove that: \[ 5BC^2 = 4(BP^2 + CQ^2) \]

7. In triangle ABC, angle A is obtuse. Given: \[ \sec(B + C) = 2 \quad \text{and} \quad \sin(2B - C) = \frac{1}{2} \] Find the measures of angles A, B, and C.

8. If \(a, b, c,\) and \(d\) are in continued geometric proportion, then prove that \[ (b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2 \]

9. If \( \alpha \) and \( \beta \) are complementary angles, then prove that \( \sin^2 \alpha + \sin^2 \beta = 1 \).

10. A and B are two complementary angles. Prove that \[ (\sin A + \sin B)^2 = 1 + 2\sin A \cos A \]

11. If \[ \sin\theta + \cos\theta = \frac{7}{5} \quad \text{and} \quad \sin\theta \cos\theta = \frac{12}{25}, \] then show that \( \sin\theta = ? \)

12. If \(\angle P + \angle Q = 90^\circ\), then **prove** that: \[ \sqrt{\frac{\sin P}{\cos Q} - \sin P \cos Q} = \cos P \]

13. If \( \sin \alpha = m \sin \beta \) and \( \tan \alpha = n \tan \beta \), then prove that \[ \cos^2 \alpha = \cfrac{m^2 - 1}{n^2 - 1} \]

14. If \( \sin x = m \sin y \) and \( \tan x = n \tan y \), then prove that \[ \cos^2 x = \cfrac{m^2 - 1}{n^2 - 1} \]