Practice Questions (Trigonometry)
1. Simplify: \( \cos^4 A - \sin^4 A \).
\((\cos^2A - \sin^2A)(\cos^2A + \sin^2A) = \cos 2A\). Answer: \( \cos 2A \).
2. If \( \triangle ABC \) is right-angled at \(C\), find \( \cos(A+B) \).
Since \(A+B=90^\circ\), \(\cos(A+B)=0\).
3. If \( \sin A + \cos A = \sqrt{2}\cos A \), find \( \tan A \).
\(\tan A = \sqrt{2}-1\).
4. If \(5\tan\theta = 4\), evaluate \( \dfrac{5\sin\theta-3\cos\theta}{5\sin\theta+3\cos\theta} \).
Result = \( \tfrac{1}{7} \).
5. If \(4\tan\theta = 3\), evaluate \( \dfrac{4\sin\theta-\cos\theta}{4\sin\theta+\cos\theta} \).
Result = \( \tfrac12 \).
6. If \( \csc A = \tfrac{13}{12} \), evaluate \( \dfrac{2\sin A-3\cos A}{4\sin A-9\cos A} \).
Result = \(3\).
7. In \(\triangle ABC\), right at \(B\), \(AB=5\), \(\sin C=\tfrac12\). Find \(AC\).
Answer: \(10\) cm.
8. Right-angled at \(C\), if \(\tan A=1\), find \(2\sin A\cos A\).
Answer: \(1\).
9. If \(\cot 2\theta=\tfrac13\), find \(\sin 3\theta\) (\(3\theta\le90^\circ\)).
Result: \(\sin 3\theta\approx0.955\).
10. If \(\tan\theta=1\), find \(\sec\theta+\csc\theta\).
Answer: \(2\sqrt2\).
11. If \(\tan\theta+\cot\theta=2\), find \(\sin^3\theta+\cos^3\theta\).
Answer: \(\tfrac{\sqrt2}{2}\).
12. In \(\triangle ABC\), right at \(B\), \(\sin A=\tfrac{7}{25}\). Find \(\cos C\).
Answer: \(\tfrac{7}{25}\).
13. Find \( \sin 45^\circ + \cos 45^\circ \).
Answer: \(\sqrt2\).
14. If \(\sin\theta=\tfrac{m}{n}\), find \(\cos\theta\).
Answer: \(\tfrac{\sqrt{n^2-m^2}}{n}\).
15. If \(\cos A=\tfrac45\), find \(\tan A\).
Answer: \(\tfrac34\).
16. In right triangle at \(B\), \(\tan A=\sqrt3\). Find \(\cos A \cos C - \sin A \sin C\).
Answer: \(0\).
17. If \(2\sin^2\beta - \cos^2\beta = 2\), find \(\beta\).
Answer: \(90^\circ\).
18. If \(\sqrt3 \tan\theta=1\), find \(\sin^2\theta-\cos^2\theta\).
Answer: \(\tfrac{\sqrt3}{2}-\tfrac12\).
19. If \(\tan A=1\), verify \(2\sin A \cos A=1\).
Verified, both sides = 1.
20. If \(\sin 2A=\tfrac12 \tan^2 45^\circ\), find \(A\).
Answer: \(15^\circ\).
21. If \(\sec A=\tfrac{15}{7}, A+B=90^\circ\), find \(\csc B\).
Answer: \(\tfrac{15}{7}\).
22. Solve: \(\sin(A+2B)=\tfrac{\sqrt3}{2}, \cos(A+B)=\tfrac12\).
Solutions: \((60^\circ,0^\circ)\) or \((0^\circ,60^\circ)\).
23. If \((1+\cos A)(1-\cos A)=\tfrac34\), find \(\tan A\).
Answer: \(\sqrt3\).
24. Evaluate \(3\cos^2 60^\circ \sec^2 30^\circ - 2\sin^2 30^\circ \tan^2 60^\circ\).
Answer: \(-\tfrac12\).
25. Solve: \(\dfrac{\cos\theta-\sin\theta}{\cos\theta+\sin\theta}=\dfrac{1-\tan\theta}{1+\tan\theta}\). Answer: Always true by dividing numerator and denominator by \(\cos\theta\).
26. If \(\tan\theta=\tfrac34\), evaluate \(\dfrac{(1+\sin\theta)(1-\sin\theta)}{(1+\cos\theta)(1-\cos\theta)}\).
Simplify: \(\dfrac{1-\sin^2\theta}{1-\cos^2\theta}=\dfrac{\cos^2\theta}{\sin^2\theta}=\cot^2\theta=(\tfrac43)^2=\tfrac{16}{9}.\)
27. Given \(\sin(A-B)=\tfrac12,\ \cos(A+B)=\tfrac12,\ 0B\). Solution: \(A=45^\circ, B=15^\circ\).
28. If \(\sin(A+B)=1, \cos(A-B)=\tfrac{\sqrt3}{2}\), solve.
Solution: \(A=60^\circ, B=30^\circ\).
29. If \(\tan(A+B)=3,\ \tan(A-B)=\tfrac13\), solve.
Solution: \(A\approx45^\circ,\ B\approx26.6^\circ\).
30. If \(\sin(A+B)=1,\ \sin(A-B)=\tfrac12\), solve.
Solution: \(A=60^\circ,\ B=30^\circ\).
31. Simplify (expression missing in scan). General method: reduce to sines/cosines and simplify.
32. If \(\theta=45^\circ\), evaluate \(2\sec^2\theta+3\csc^2\theta\).
Answer: \(10\).
33. Prove: \(\dfrac{1+\sin A}{1-\sin A}= \left(\dfrac{\sec A+1}{\sec A-1}\right)^2\).
Verified using conjugates and \(\sec A=\tfrac1{\cos A}\).
34. Prove: \((1+\tan^2\theta)(1+\sin\theta)(1-\sin\theta)=1\).
LHS = \(\sec^2\theta\cdot\cos^2\theta=1\).
35. Prove: \(\dfrac{\cos\theta}{\sin\theta}+\dfrac{\sin\theta}{\cos\theta}= \sec\theta\csc\theta\).
LHS = \(\dfrac{\cos^2\theta+\sin^2\theta}{\sin\theta\cos\theta}= \dfrac{1}{\sin\theta\cos\theta}= \csc\theta\sec\theta\).
36. Prove: \((\sin A+\csc A)^2+(\cos A+\sec A)^2=7+\tan^2A+\cot^2A\).
Expand both sides, use \(\csc^2=1+\cot^2\), \(\sec^2=1+\tan^2\), simplify → equality holds.
37. Prove: \(\dfrac{\cos A+1}{\sin A}=\dfrac{\sec A+\cos A}{1+\sin A}\).
Cross-multiply and simplify with \(\sec A=1/\cos A\). Verified.
38. Prove: \(\dfrac{\sin\theta\cos\theta-1}{\sin\theta\cos\theta+1}=\dfrac{\sec\theta+\tan\theta}{\sec\theta-\tan\theta}\).
Expand RHS → simplify → matches LHS. Verified.
39. Prove: \((1+\cot\theta-\csc\theta)(1+\tan\theta+\sec\theta)=2\).
Expand and use identities, simplifies to 2. Verified.
40. Prove: \(\dfrac{\cot A-\cos A}{\cot A+\cos A}=\dfrac{\csc A-1}{\csc A+1}\).
Multiply top & bottom by \(\sin A\). Simplify → equality holds.
41. Prove: \(\dfrac{\cos^2\theta+\sin^2\theta}{1-\tan\theta- \cot\theta} = 1\).
Simplify using Pythagorean identities. Verified.
42. If \(\cos\theta+\sin\theta=\sqrt2\cos\theta\), show \(\cos\theta-\sin\theta=\sqrt2\sin\theta\).
Subtract equation from \(\sqrt2(\sin\theta+\cos\theta)\). Verified.
43. Prove: \(\dfrac{\tan\theta+\cot\theta}{1-\cot\theta-\tan\theta}=\sec\theta+\cos\theta\).
Simplify both sides to \(\dfrac{1}{\sin\theta\cos\theta}\). Verified.
44. Prove: \(\dfrac{\cos\theta-\sin\theta}{\cos\theta+\sin\theta}=\dfrac{\cos\theta\cot\theta+1}{\cos\theta+\sin\theta}\).
Reduce both to same form with identities. Verified.
45. Prove: \(\dfrac{1+\tan^2A}{1-\cot A}=\tan A\).
Simplify LHS: \(\dfrac{\sec^2A}{1-\cos A/\sin A} = \tan A\). Verified.
