Class 12 inverse trigonometry important questions: Your Gateway to Excellence
Class 12 inverse trigonometry important questions demand focused practice. Principal values, properties, and complex simplifications often appear in board papers. Regular revision builds accuracy. Short practice sessions yield better retention. Mastering these questions creates a strong foundation for calculus. Use quality resources to stay ahead. Consistency is key.
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Inverse Trigonometry HOTS & Important Questions Class 12 Mathematics
1. Prove that:
\[ \tan\left( \frac{\pi}{4} + \frac{1}{2} \cos^{-1} \frac{a}{b} \right) + \tan\left( \frac{\pi}{4} – \frac{1}{2} \cos^{-1} \frac{a}{b} \right) = \frac{2b}{a} \]2. If \( \cos^{-1} x + \cos^{-1} y + \cos^{-1} z = \pi \), prove that \( x^2 + y^2 + z^2 + 2xyz = 1 \).
3. Solve for \( x \):
\[ \sin^{-1} \frac{5}{x} + \sin^{-1} \frac{12}{x} = \frac{\pi}{2} \]4. Prove that:
\[ \tan^{-1} \left( \frac{\sqrt{1+x^2} + \sqrt{1-x^2}}{\sqrt{1+x^2} – \sqrt{1-x^2}} \right) = \frac{\pi}{4} + \frac{1}{2} \cos^{-1} x^2, \quad -1 < x < 1 \]5. If \( \sin(\sin^{-1} \frac{1}{5} + \cos^{-1} x) = 1 \), then find the value of \( x \).
6. Simplify the expression:
\[ \tan^{-1} \left( \frac{3a^2x – x^3}{a(a^2 – 3x^2)} \right), \quad a > 0, \quad \frac{-a}{\sqrt{3}} < x < \frac{a}{\sqrt{3}} \]7. If \( \cos^{-1} \frac{x}{a} + \cos^{-1} \frac{y}{b} = \alpha \), prove that:
\[ \frac{x^2}{a^2} – \frac{2xy}{ab} \cos \alpha + \frac{y^2}{b^2} = \sin^2 \alpha \]8. Solve the equation for \( x \):
\[ \tan^{-1} (2x) + \tan^{-1} (3x) = \frac{\pi}{4} \]9. Prove that:
\[ 2 \tan^{-1} \left( \tan \frac{\alpha}{2} \tan \left( \frac{\pi}{4} – \frac{\beta}{2} \right) \right) = \tan^{-1} \left( \frac{\sin \alpha \cos \beta}{\cos \alpha + \sin \beta} \right) \]10. Find the simplest form of:
\[ \sin^{-1} \left( \frac{\sin x + \cos x}{\sqrt{2}} \right), \quad -\frac{\pi}{4} < x < \frac{\pi}{4} \]11. If \( ( \tan^{-1} x )^2 + ( \cot^{-1} x )^2 = \frac{5\pi^2}{8} \), then find \( x \).
12. Prove the following identity:
\[ \tan^{-1} \left( \frac{1}{1+1 \cdot 2} \right) + \tan^{-1} \left( \frac{1}{1+2 \cdot 3} \right) + \] \[ \dots + \tan^{-1} \left( \frac{1}{1+n(n+1)} \right) = \tan^{-1} (n+1) – \frac{\pi}{4} \]13. Solve for \( x \):
\[ \sin \left[ 2 \cos^{-1} \left\{ \cot (2 \tan^{-1} x) \right\} \right] = 0 \]14. Find the value of:
\[ \cos \left( \sec^{-1} x + \csc^{-1} x \right), \quad |x| \ge 1 \]15. If \( a > b > c > 0 \), prove that:
\[ \cot^{-1} \left( \frac{ab+1}{a-b} \right) + \cot^{-1} \left( \frac{bc+1}{b-c} \right) + \cot^{-1} \left( \frac{ca+1}{c-a} \right) = \pi \]Answers and Hints
1. Hint: Use the identity for \(\tan(A+B)\) and let \(\theta = \frac{1}{2} \cos^{-1} \frac{a}{b}\).
2. Hint: Use the identity for the sum of three \(\cos^{-1}\) terms and square both sides.
3. Answer: \(x = 13\). Hint: Use \(\sin^{-1} A + \sin^{-1} B = \frac{\pi}{2} \Rightarrow A^2 + B^2 = 1\).
4. Hint: Let \(x^2 = \cos 2\theta\) and simplify the expression inside \(\tan^{-1}\).
5. Answer: \(x = \frac{1}{5}\). Hint: \(\sin^{-1} \frac{1}{5} + \cos^{-1} x = \frac{\pi}{2}\).
6. Answer: \(3 \tan^{-1} \frac{x}{a}\). Hint: Let \(x = a \tan \theta\) and simplify.
7. Hint: Use the identity for \(\cos(\alpha – \beta)\) and express in terms of \(x, y, a, b\).
8. Answer: \(x = \frac{1}{6}\) (Reject \(x = -1\)). Hint: Use \(\tan^{-1} A + \tan^{-1} B = \tan^{-1} \left( \frac{A+B}{1-AB} \right)\).
9. Hint: Use the formula for \(\tan^{-1} \left( \tan \frac{A}{2} \tan \frac{B}{2} \right)\) and simplify.
10. Answer: \(x + \frac{\pi}{4}\). Hint: Rewrite \(\sin x + \cos x\) as \(\sqrt{2} \sin \left( x + \frac{\pi}{4} \right)\).
11. Answer: \(x = -1\). Hint: Use \((\tan^{-1} x) + (\cot^{-1} x) = \frac{\pi}{2}\) and solve.
12. Hint: Use telescoping series and the identity \(\tan^{-1} \left( \frac{1}{1+k(k+1)} \right) = \tan^{-1} (k+1) – \tan^{-1} k\).
13. Answer: \(x = \pm 1, \pm(\sqrt{2}-1), \pm(\sqrt{2}+1)\). Hint: Simplify \(\cot(2 \tan^{-1} x)\) and solve \(\sin(2 \cos^{-1} y) = 0\).
14. Answer: \(0\). Hint: \(\sec^{-1} x + \csc^{-1} x = \frac{\pi}{2}\).
15. Hint: Use the identity for \(\cot^{-1} A + \cot^{-1} B\) and simplify.
Frequently Asked Questions (FAQs)
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