1. If \[ \tan \theta \cdot \cos 60^\circ = \frac{\sqrt{3}}{2} \] then find the value of \[ \sin(\theta - 15^\circ) \] given that \(0^\circ < \theta < 90^\circ\).
2. If \(∠P + ∠Q = 90^\circ\), then prove that \[ \sqrt{\frac{\sin P}{\cos Q}} - \sin P \cos Q = \cos^2 P \]
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. Evaluate the expression: \[ \frac{1 - \sin^2 30^\circ}{1 + \sin^2 45^\circ} \times \frac{\cos^2 60^\circ + \cos^2 30^\circ}{\csc^2 90^\circ - \cot^2 90^\circ} \div (\sin 60^\circ \tan 30^\circ) \]
5. Evaluate the expression: \[ \frac{(\sin 0^\circ + \sin 60^\circ)(\cos 60^\circ + \cot 45^\circ)}{(\cot 60^\circ + \tan 30^\circ)(\csc 30^\circ - \csc 90^\circ)} \]
6. If \[ x \cos 60^\circ = \frac{2 \tan 45^\circ}{1 + \tan^2 45^\circ} - \frac{1 - \tan^2 30^\circ}{1 + \tan^2 30^\circ} \] then find the value of \(x\).
7. Show that: \[ \sqrt{\cfrac{1 + \cos 30^\circ}{1 - \cos 30^\circ}} = \sec 60^\circ + \tan 60^\circ \]
8. \[ \tan 60^\circ = \sqrt{3},\quad \tan 30^\circ = \frac{1}{\sqrt{3}} \\ \cos 60^\circ = \frac{1}{2},\quad \cos 30^\circ = \frac{\sqrt{3}}{2} \\ \sin 60^\circ = \frac{\sqrt{3}}{2},\quad \sin 30^\circ = \frac{1}{2} \] \[ \frac{\tan 60^\circ - \tan 30^\circ}{1 + \tan 60^\circ \cdot \tan 30^\circ} = \frac{\sqrt{3} - \frac{1}{\sqrt{3}}}{1 + \sqrt{3} \cdot \frac{1}{\sqrt{3}}} = \frac{\frac{3 - 1}{\sqrt{3}}}{2} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}} \] \[ \cos 60^\circ \cos 30^\circ + \sin 60^\circ \sin 30^\circ = \frac{1}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{1}{2} = \frac{\sqrt{3}}{4} + \frac{\sqrt{3}}{4} = \frac{2\sqrt{3}}{4} = \frac{\sqrt{3}}{2} \] \[ \frac{1}{\sqrt{3}} + \frac{\sqrt{3}}{2} = \frac{2 + 3}{2\sqrt{3}} = \frac{5}{2\sqrt{3}} \]
9. If \(\angle P + \angle Q = 90^\circ\), then **prove** that: \[ \sqrt{\frac{\sin P}{\cos Q} - \sin P \cos Q} = \cos P \]
10. \[ \sin 43^\circ \cos 47^\circ + \cos 43^\circ \sin 47^\circ = \sin(43^\circ + 47^\circ) = \sin 90^\circ = 1 \]
(a) 0 (b) 4 (c) sin4° (d) 1
11. If \( \tan \theta = \frac{x}{y} \), then what is the value of \[ \frac{x\sin\theta - y\cos\theta}{x\sin\theta + y\cos\theta}? \]
(a) \(\cfrac{x^2+y^2}{x^2-y^2}\) (b) \(\cfrac{x-y}{x+y}\) (c) \(\cfrac{x+y}{x-y}\) (d) \(\cfrac{x^2-y^2}{x^2+y^2}\)
12. From the equation \( \sin(90^\circ + \theta) = \cos(120^\circ - 3\theta) \), the value of \( \theta \) is —
(a) 30° (b) 60° (c) 45° (d) None of the above
13. If A = B = 30°, show that \[ \sin A \cos B + \cos A \sin B = \sin(A + B) \]
14. If \( \cot\theta = \frac{x}{y} \), then prove that \[ \frac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta} = \frac{x^2 - y^2}{x^2 + y^2} \]
15. If \[ \frac{\sinθ}{x} = \frac{\cosθ}{y} \] then prove that \[ \sinθ - \cosθ = \frac{x - y}{\sqrt{x^2 + y^2}} \]
16. Evaluate the expression: \[ \cot^2 30^\circ - 2\cos^2 60^\circ - 4\sin^2 30^\circ - \frac{3}{4}\sec^2 45^\circ + \tan 45^\circ \]
17. In triangle \(\triangle ABC\), prove that: \[ \sin\left(\frac{A + B}{2}\right) + \cos\left(\frac{B + C}{2}\right) = \cos\left(\frac{C}{2}\right) + \sin\left(\frac{A}{2}\right) \]
18. Prove that \[ \sqrt{\cfrac{1 + \cos \theta}{1 - \cos \theta}} = \csc \theta + \cot \theta \]
19. If \[ \frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = 3 \] then find the value of \[ \sin^4 \theta - \cos^4 \theta \]
20. Given: \[ x \cos \theta = 3 \quad \text{and} \quad 4 \tan \theta = y \] Find the relation between \(x\) and \(y\) eliminating \(\theta\).
21. If \(\tan\theta = \frac{x}{y}\), then what is the value of \[ \frac{x\sin\theta - y\cos\theta}{x\sin\theta + y\cos\theta}? \]
(a) \(\cfrac{x^2-y^2}{x^2+y^2}\) (b) \(\cfrac{y^2-x^2}{x^2+y^2}\) (c) \(\cfrac{x^2+y^2}{y^2-x^2}\) (d) None of the above
22. If \(\tan\theta = \cos 30^\circ + \sin 60^\circ\), then \(\sin(90^\circ - \theta)\) will be –
(a) \(\cfrac{1}{3}\) (b) \(\cfrac{1}{4}\) (c) \(\cfrac{1}{2}\) (d) \(1\)
23. If \( \tan\theta = \frac{x}{y} \), then find the value of \[ \frac{x \sin\theta - y \cos\theta}{x \sin\theta + y \cos\theta} \]
24. If \[ x \cot\left(\frac{\pi}{6}\right) = 2 \cos\left(\frac{\pi}{3}\right) + \frac{3}{4} \sec^2\left(\frac{\pi}{4}\right) + 4 \sin\left(\frac{\pi}{6}\right) \] then find the value of \(x\).
25. Prove that: \(\sec 70^\circ \sin 20^\circ + \cos 20^\circ \csc 70^\circ = 2\)
26. If \( \alpha \) and \( \beta \) are complementary angles, then prove that \[ \cot \beta + \cos \beta = \frac{\cos \beta}{\cos \alpha (1 + \sin \beta)} \]
27. If \( \alpha \) and \( \beta \) are complementary angles, then prove that \[ \frac{\sec \alpha}{\cos \alpha} - \cot^2 \beta = 1 \]
28. If ∠P + ∠Q = 90°, then show that \[ \sqrt{\frac{\sin P}{\cos Q} - \sin P \cos Q} = \cos P \]
29. If \(\cot A = \frac{4}{7.5}\), then find the values of \(\cos A\) and \(\csc A\), and show that: \[ 1 + \cot^2 A = \csc^2 A \]
30. Prove that \[ \frac{\tan \theta + \sec \theta - 1}{\tan \theta - \sec \theta + 1} = \frac{1 + \sin \theta}{\cos \theta} \]
