CBSE Class 12 Inverse Trigonometric Functions PYQs Analysis (2021-2025)

Based on detailed analysis of last 5 years' papers. Perfect for 2026 Boards prep!

Key Trends

Weightage Breakdown: 1-2 MCQs/assertion-reason (1 mark), 1 short answer/simplification (2-3 marks), occasional 4-mark question on identities/proofs.
Repeated Topics: Principal value of inverse trig functions, simplification of expressions like tan⁻¹x + tan⁻¹y, sin⁻¹x + cos⁻¹x = π/2, 2tan⁻¹x = tan⁻¹(...), range/domain checks.
Difficulty: Easy to medium; direct formula application or algebraic simplification key.
Changes: More MCQs and assertion-reason in recent years; focus on exact value calculations.

Most Important and Repeated Questions

Question Example	Type/Marks	Years Repeated	Notes
Find the principal value of sin⁻¹(1/2) + cos⁻¹(-1/√2).	Short Answer (2 marks)	2021, 2022, 2023, 2024, 2025	Repeated 5x; Use principal values: π/6 + 3π/4 = 11π/12.
Prove that sin⁻¹x + cos⁻¹x = π/2 for x ∈ [-1,1].	Proof (3-4 marks)	2021 Term 1, 2022, 2023, 2025	Repeated 4x; Let θ = sin⁻¹x → cos⁻¹x = π/2 - θ.
Simplify: tan⁻¹(1/√3) + tan⁻¹(1) + tan⁻¹(√3).	Short Answer (2 marks)	2022, 2023, 2024, 2025	Repeated 4x; π/6 + π/4 + π/3 = 11π/12 (or π - π/12 if >π/2 check).
Assertion: The range of tan⁻¹x is (-π/2, π/2). Reason: tan is bijective in this interval.	Assertion-Reason (1 mark)	2023, 2024, 2025	Repeated 3x; Both true, reason correct explanation.
Find the value of tan⁻¹(3) - tan⁻¹(1/3).	Short Answer (2 marks)	2021 Term 2, 2022, 2024	Repeated 3x; Use formula tan⁻¹a - tan⁻¹b = tan⁻¹((a-b)/(1+ab)) = π/4.
Evaluate cos⁻¹(-1/2) + sin⁻¹(√3/2).	Short Answer (2 marks)	2023, 2025	Repeated 2x; 2π/3 + π/3 = π.
MCQ: Principal value of cos⁻¹(-1/√2) is: (a) π/4 (b) 3π/4 (c) -π/4 (d) 5π/4	MCQ (1 mark)	2021 Term 1, 2024	Repeated 2x; Answer (b) 3π/4.
Simplify: 2 tan⁻¹(1/√5).	Short Answer (2 marks)	2022, 2025	Repeated 2x; tan⁻¹(4/3) using double angle formula.
Case-based: Given expression involving inverse tan, find value or prove identity.	Case-Based (4 marks)	2023, 2025	Repeated 2x; Focus on algebraic manipulation.
Find principal value of tan⁻¹(-1/√3).	Short Answer (1-2 marks)	2021 Term 1, 2023	Repeated 2x; -π/6 (principal range (-π/2, π/2)).

Best Format for Preparation and Answering

Study Format: Memorize principal ranges & values table. Practice simplification using identities (tan⁻¹a + tan⁻¹b formula, etc.). Solve 10–15 PYQs daily.
Answer Format: Write exact value + steps (e.g., use identity or direct principal value). For proofs: Let θ = inverse function → apply trig identity → conclude.
Tips: Focus on tan⁻¹ sums and principal values (high repeat). Time: 4–6 mins per question. Very scoring chapter if formulas are clear.

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