Chapter 2 · Class XII · NCERT Exercises

Inverse Trigonometric Functions — Exercises

Principal Values to Proof-Based Identities — Complete Inverse Trig Solutions

📂 3 Exercises📝 66 Questions

▶ Start Exercise 2.1 📖 Notes

Exercise Index

①

Exercise 2.1

Principal value branches; domain and range of inverse trig functions

14 QModerate

→

②

Exercise 2.2

Properties of inverse trig functions; identity proofs

35 QHard

→

★

Miscellaneous

Combined identity proofs; equations in inverse trig form

17 QHard

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3 files · 66 questions

Chapter Snapshot

JEE MainCBSE BoardsBITSAT

8Concepts

18Formulas

Moderate-HighDifficulty

5–6%Weightage

Before You Begin

Prerequisites

✓Class XI — Trigonometric Functions (Ch 3)
✓Unit circle and quadrant sign rules
✓Function inverses (Ch 1)

Have Ready

🔧Principal value branch table
🔧Unit circle diagram

Syllabus-wise Topic Map

2.1 IntroductionWhy trig functions need restricted domains to be invertible

2.2 Basic ConceptsPrincipal value branches of sin⁻¹, cos⁻¹, tan⁻¹, etc.; evaluating standard values

2.3 Properties of Inverse Trigonometric Functionssin⁻¹x+cos⁻¹x=π/2 type identities; sum/difference formulas; conversions between inverse ratios

Key Formulae

\(\sin^{-1}x + \cos^{-1}x = \dfrac{\pi}{2}\)

\(\tan^{-1}x + \cot^{-1}x = \dfrac{\pi}{2}\)

\(\tan^{-1}x + \tan^{-1}y = \tan^{-1}\!\left(\dfrac{x+y}{1-xy}\right),\ xy<1\)

\(2\tan^{-1}x = \sin^{-1}\!\left(\dfrac{2x}{1+x^2}\right),\ |x|\leq 1\)

\(\text{Range of }\sin^{-1}x: \left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]\)

NCERT Solving Method

Step 1 — Always confirm the value lies within the principal value branch before writing the final answer. Step 2 — For proofs: substitute x=tanθ or x=sinθ to simplify surds under inverse trig. Step 3 — Use complementary identities (sin⁻¹+cos⁻¹=π/2) to convert between ratios. Step 4 — For sum formulas, check the xy<1 or xy>1 condition before applying the tan⁻¹ addition rule.