1. Show that: \[ \cfrac{2 \tan^2 30^\circ}{1 - \tan^2 30^\circ} + \sec^2 45^\circ - \cot^2 45^\circ = \sec 60^\circ \]

2. Show that \[ \frac{2 \tan^2 30^\circ}{1 - \tan^2 30^\circ} + \sec^2 45^\circ - \cot^2 45^\circ = \sec 60^\circ \]

3. Left-hand side (LHS): \[ 1 + \cfrac{\tan A}{\tan B} = 1 + \cfrac{\tan(90^\circ - B)}{\tan B} = 1 + \cfrac{\cot B}{\cot B} = 1 + \cot^2 B = \csc^2 B \] Right-hand side (RHS): \[ \tan^2 A \cdot \sec^2 B = \tan^2(90^\circ - B) \cdot \sec^2 B = \cot^2 B \cdot \sec^2 B = \cfrac{\cos^2 B}{\sin^2 B} \cdot \cfrac{1}{\cos^2 B} = \cfrac{1}{\sin^2 B} = \csc^2 B \] \(\therefore\) LHS = RHS (Proved)

4. If \(\cot θ = 2\), then find the values of \(\tan θ\) and \(\sec θ\), and show that: \[1 + \tan^2θ = \sec^2θ\]

5. If \(\cot A = \frac{4}{7.5}\), then find the values of \(\cos A\) and \(\csc A\), and show that: \[ 1 + \cot^2 A = \csc^2 A \]

6. If \( \tan 9^\circ = \frac{a}{b} \), then prove that \[ \frac{\sec^2 81^\circ}{1 + \cot^2 81^\circ} = \frac{b^2}{a^2} \]

7. What is the value of \[ \frac{4}{\sec^2θ} + \frac{1}{1 + \cot^2θ} + 3\sin^2θ? \]

8. 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: \[ S = 2V\left(\frac{1}{h} + \frac{1}{r}\right) \]

9. Evaluate: \[ 3\tan^2 45^\circ - \sin^2 60^\circ - \frac{1}{3} \cot^2 30^\circ - \frac{1}{8} \sec^2 45^\circ \]

10. If AB and AC are chords of the larger of two concentric circles, and they touch the smaller circle at points P and Q respectively, prove that: \[ PQ = \frac{1}{2}BC \]

11. Given: AC is the diameter of a circle centered at O, ABC is an inscribed triangle, and OP ⊥ AB (where P lies on the circle). **Prove that:** \[ OP : BC = 1 : 2 \]

12. If \(\angle A + \angle B = 90^\circ\), then prove that \[ 1 + \cfrac{\tan A}{\tan B} = \tan^2 A \sec^2 B \]

13. Solve: \[ x^2 = \sin^2 30^\circ + 4\cot^2 45^\circ - \sec^2 60^\circ \]

14. Find the value of: \[ \frac{2\tan^2 30^\circ}{1 - \tan^2 30^\circ} + \sec^2 45^\circ - \cot^2 45^\circ - \sec 60^\circ \]

15. Given: \[ x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1 \] Show that: \[ \frac{a}{a + x} + \frac{b}{b + y} + \frac{c}{c + z} = 1 \]

16. If \(a + b + c = 0\), then prove that: \[ \cfrac{1}{2a^2 + bc} + \cfrac{1}{2b^2 + ca} + \cfrac{1}{2c^2 + ab} = 0 \]

17. If \( \alpha \) and \( \beta \) are complementary angles, then prove that \[ \frac{\sec \alpha}{\cos \alpha} - \cot^2 \beta = 1 \]

18. If ∠A + ∠B = 90°, then prove that \[ 1 + \frac{\tan A}{\tan B} = \sec^2 A \]

19. What is the value of \[ \left(\cfrac{4}{\sec^2 \theta}+\cfrac{1}{1+\cot^2 \theta}+3 \sin^2 \theta \right) \]

20. Show that: \[ \sqrt{\cfrac{1 + \cos 30^\circ}{1 - \cos 30^\circ}} = \sec 60^\circ + \tan 60^\circ \]

21. Show that: \[ \tan^2 \left( \cfrac{\pi}{4} \right) \cdot \sin \left( \cfrac{\pi}{3} \right) \cdot \tan \left( \cfrac{\pi}{6} \right) \cdot \tan^2 \left( \cfrac{\pi}{3} \right) = 1 \cfrac{1}{2} \]

22. "P and Q are two points on a straight line. From points P and Q, perpendiculars PR and QS are drawn respectively. PS and QR intersect at point O. OT is drawn perpendicular to PQ. Prove that: \[ \frac{1}{OT}=\frac{1}{PR}+\frac{1}{QS} \]"

23. \[ \frac{a^2}{b + c} = \frac{b^2}{c + a} = \frac{c^2}{a + b} = 1 \] **Show that:** \[ \frac{1}{1 + a} = \frac{1}{1 + b} = \frac{1}{1 + c} = 1 \]

24. Determine the value of \[ \sec^2 60^\circ - \cot^2 30^\circ - \frac{2 \tan 30^\circ \cdot \csc 60^\circ}{1 + \tan^2 30^\circ} \]

25. If a planet's gravitational force on its satellite is directly proportional to the planet's mass (M) and inversely proportional to the square of their distance (D), and the square of the satellite's orbital period (T) is directly proportional to the cube of the distance and inversely proportional to the gravitational force, then for two sets of values \(m_1, d_1, t_1\) and \(m_2, d_2, t_2\) corresponding to M, D, and T respectively, prove that: \[ m_1 t_1^2 d_2^3 = m_2 t_2^2 d_1^3 \]

26. If \( \sin 17^\circ = \frac{x}{y} \), then prove that: \[ \sec 17^\circ - \sin 73^\circ = \frac{x^2}{y\sqrt{y^2 - x^2}} \]

27. A person bought \(y\) pencils for ₹\(x\). If each pencil had cost ₹1 less, then with the same amount ₹\(x\), he would have received one additional pencil. Prove that: \[ 2y = \sqrt{1 + 4x} - 1 \]

28. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that: \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

29. Evaluate the following expression: \[ \sec^2 60^\circ - \cot^2 30^\circ - \frac{2 \tan 30^\circ \csc 60^\circ}{1 + \tan^2 30^\circ} \]

30. Find the value of > \[ \left(\frac{4}{\sec^2\theta} + \frac{1}{1 + \cot^2\theta} + 3\sin^2\theta\right) \]