SECTION C — Short Answer Type Questions [3 Marks Each]
- 3 Marks Q26. If $A, B, C$ are interior angles of a triangle $ABC$, prove that $\tan\left(\frac{A + B}{2}\right) = \cot\left(\frac{C}{2}\right)$.
- 3 Marks Q27. Evaluate: $\frac{\cos^2 20^\circ + \cos^2 70^\circ}{\sec^2 50^\circ - \cot^2 40^\circ} + 2\csc^2 58^\circ - 2\cot 58^\circ \tan 32^\circ - 4\tan 13^\circ \tan 37^\circ \tan 77^\circ \tan 53^\circ$.
- 3 Marks Q28. If $\sec 5A = \csc(A - 36^\circ)$, where $5A$ is an acute angle, find the value of $A$.
- 3 Marks Q29. Simplify and find the value: $\frac{\sin 27^\circ}{\cos 63^\circ} + \frac{\cos 63^\circ}{\sin 27^\circ}$
- 3 Marks Q30. If $\cos \theta = \sin\left(\theta - 45^\circ\right)$, where $\theta$ and $\theta - 45^\circ$ are acute angles, find the measure of $\theta$.
- 3 Marks Q31. Prove the identity: $\frac{\sin \theta - 2\sin^3 \theta}{2\cos^3 \theta - \cos \theta} = \tan \theta$.
- 3 Marks Q32. Prove that: $(\sin \theta + \csc \theta)^2 + (\cos \theta + \sec \theta)^2 = 7 + \tan^2 \theta + \cot^2 \theta$.
- 3 Marks Q33. Simplify the expression: $\frac{1}{\csc \theta - \cot \theta} - \frac{1}{\sin \theta} = \frac{1}{\sin \theta} - \frac{1}{\csc \theta + \cot \theta}$.
- 3 Marks Q34. Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \sec \theta \csc \theta$.
- 3 Marks Q35. Prove that: $(1 + \cot A - \csc A)(1 + \tan A + \sec A) = 2$.
- 3 Marks Q36. Prove the identity: $\frac{\cos A}{1 - \tan A} + \frac{\sin^2 A}{\sin A - \cos A} = \sin A + \cos A$.
- 3 Marks Q37. If $\sin \theta + \cos \theta = p$ and $\sec \theta + \csc \theta = q$, show that $q(p^2 - 1) = 2p$.
- 3 Marks Q38. Prove that: $\sqrt{\frac{1 + \cos A}{1 - \cos A}} + \sqrt{\frac{1 - \cos A}{1 + \cos A}} = 2\csc A$.
- 3 Marks Q39. Prove that: $\frac{\sin A - 2\sin^3 A}{2\cos^3 A - \cos A} = \tan A$.
- 3 Marks Q40. If $\frac{cos \alpha}{\cos \beta} = m$ and $\frac{\cos \alpha}{\sin \beta} = n$, prove that $(m^2 + n^2)\cos^2 \beta = n^2$.
- 3 Marks Q41. If $x = a \sec \theta + b \tan \theta$ and $y = a \tan \theta + b \sec \theta$, prove that $x^2 - y^2 = a^2 - b^2$.
- 3 Marks Q42. If $a \cos \theta - b \sin \theta = c$, prove that $a \sin \theta + b \cos \theta = \pm \sqrt{a^2 + b^2 - c^2}$.
- 3 Marks Q43. If $\tan \theta + \sin \theta = m$ and $\tan \theta - \sin \theta = n$, prove that $m^2 - n^2 = 4\sqrt{mn}$.
- 3 Marks Q44. If $\sec \theta + \tan \theta = p$, show that $\frac{p^2 - 1}{p^2 + 1} = \sin \theta$
- 3 Marks Q45. If $sin \theta + 2\cos \theta = 1$, then prove that $2\sin \theta - \cos \theta = 2$.
- 3 Marks Q46. If $a \sin \theta + b \cos \theta = c$, then prove that $a \cos \theta - b \sin \theta = \pm \sqrt{a^2 + b^2 - c^2}$.
- 3 Marks Q47. Eliminate $\theta$ from the equations: $x = r \sin \alpha \cos \theta$, $y = r \sin \alpha \sin \theta$, and $z = r \cos \alpha$.
- 3 Marks Q48. If $\cos \theta + \sin \theta = \sqrt{2}\cos \theta$, show that $\cos \theta - \sin \theta = \sqrt{2}\sin \theta$.
- 3 Marks Q49. If $\frac{\cos \theta}{1 - \sin \theta} = x$, prove that $\frac{\cos \theta}{1 + \sin \theta} = \frac{1}{x}$.
- 3 Marks Q50. If $p = \sec \theta + \tan \theta$ and $q = \csc \theta + \cot \theta$, express the product $pq$ purely in terms of $\tan \theta$ and $\sec \theta$ or simplify to a single ratio form where applicable.
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English: These model solutions are structured to global academic standards for self-assessment and practice purposes. Students are advised to verify with official textbooks for board examinations.
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