SECTION E — Long Answer Type Questions [5 Marks Each]
- 5 Marks Q1. Prove that: $\frac{\sin \theta - 2\sin^3 \theta}{2\cos^3 \theta - \cos \theta} = \tan \theta$
- 5 Marks Q2. Prove that: $(\sin A + \csc A)^2 + (\cos A + \sec A)^2 = 7 + \tan^2 A + \cot^2 A$
- 5 Marks Q3. Prove that: $\frac{\tan \theta}{1 - \cot \theta} + \frac{\cot \theta}{1 - \tan \theta} = 1 + \sec \theta \csc \theta$.
- 5 Marks Q4. Prove that: $\frac{1 + \sec A}{\sec A} = \frac{\sin^2 A}{1 - \cos A}$
- 5 Marks Q5. Prove that: $\sqrt{\frac{1 + \sin A}{1 - \sin A}} = \sec A + \tan A$.
- 5 Marks Q6. Prove the identity: $\frac{\cos A}{1 - \tan A} + \frac{\sin^2 A}{\sin A - \cos A} = \sin A + \cos A$.
- 5 Marks Q7. Prove that: $(1 + \cot A - \csc A)(1 + \tan A + \sec A) = 2$.
- 5 Marks Q8. Prove that: $\frac{1 + \tan^2 A}{1 + \cot^2 A} = \left(\frac{1 - \tan A}{1 - \cot A}\right)^2 = \tan^2 A$.
- 5 Marks Q9. Prove that: $\sqrt{\frac{1 + \cos A}{1 - \cos A}} + \sqrt{\frac{1 - \cos A}{1 + \cos A}} = 2\csc A$.
- 5 Marks Q10. If $\sin \theta + \cos \theta = p$ and $\sec \theta + \csc \theta = q$, show that $q(p^2 - 1) = 2p$.
- 5 Marks Q11. If $x = a \sec \theta + b \tan \theta$ and $y = a \tan \theta + b \sec \theta$, prove that $x^2 - y^2 = a^2 - b^2$.
- 5 Marks Q12. If $a \cos \theta - b \sin \theta = c$, prove that $a \sin \theta + b \cos \theta = \pm \sqrt{a^2 + b^2 - c^2}$
- 5 Marks Q13. If $\tan \theta + \sin \theta = m$ and $\tan \theta - \sin \theta = n$, prove that $m^2 - n^2 = 4\sqrt{mn}$.
- 5 Marks Q14. If $\sec \theta + \tan \theta = p$, show that $\frac{p^2 - 1}{p^2 + 1} = \sin \theta$.
- 5 Marks Q15. If $\frac{\cos \alpha}{\cos \beta} = m$ and $\frac{\cos \alpha}{\sin \beta} = n$, prove that $(m^2 + n^2)\cos^2 \beta = n^2$.
- 5 Marks Q16. If $p = \sec \theta + \tan \theta$ and $q = \csc \theta + \cot \theta$, express the product $pq$ in terms of basic ratios and show its relationship with $\tan \theta$ and $\sec \theta$.
- 5 Marks Q17. Evaluate the following expression completely by substituting standard trigonometric values: $$\frac{5\cos^2 60^\circ + 4\sec^2 30^\circ - \tan^2 45^\circ}{\sin^2 30^\circ + \cos^2 30^\circ}$$
- 5 Marks Q18. If $\theta = 30^\circ$, verify both sides of the triple angle identity: $$\cos 3\theta = 4\cos^3 \theta - 3\cos \theta$$
- 5 Marks Q19. Evaluate the multi-term expression without using direct trigonometric tables: $$\frac{\cos^2 20^\circ + \cos^2 70^\circ}{\sec^2 50^\circ - \cot^2 40^\circ} + 2\csc^2 58^\circ - 2\cot 58^\circ \tan 32^\circ - 4\tan 13^\circ \tan 37^\circ \tan 77^\circ \tan 53^\circ$$
- 5 Marks Q20. Given that $\sin(A + B) = \sin A \cos B + \cos A \sin B$, first find the values of acute angles $A$ and $B$ if $\tan(A + B) = \sqrt{3}$ and $\tan(A - B) = \frac{1}{\sqrt{3}}$, and then verify whether the sine addition formula holds true for these calculated values of $A$ and $B$
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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.
தமிழ்: இந்த மாதிரி கணிதத் தீர்வுகள் மாணவர்களின் சுய கற்றல் மற்றும் பயிற்சித் திறனுக்காக உலகளாவிய கல்வித் தரத்தில் உருவாக்கப்பட்டுள்ளன. தேர்வுகளுக்குத் தயாராகும் போது உங்கள் அதிகாரப்பூர்வ பாடப்புத்தகத்துடன் சரிபார்க்கவும்.
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