Class 12 Maths NCERT Solutions Chapter 2 Inverse Trigonometric Functions | Boundless Maths

Chapter 2 Class 12 Maths NCERT Solutions · Unit I · Relations and Functions

Class 12 Maths NCERT Solutions Chapter 2: Inverse Trigonometric Functions

Free, step-by-step NCERT Solutions for both exercises of this chapter — principal values of inverse trigonometric functions, and their properties and identities — plus the full Miscellaneous Exercise. Solved the way CBSE awards marks, with the key formulas, the substitution to reach for, and the mistakes that cost students marks every year, right on this page.

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Class 12 Maths NCERT Solutions Chapter 2 — Overview

This Class 12 Maths NCERT Solutions Chapter 2 hub covers Inverse Trigonometric Functions, which picks up directly from Chapter 1's discussion of invertible functions. Since sine, cosine and the other trig functions repeat their values endlessly, none of them are one-one over their full domain — so their inverses only exist once you restrict the domain to a single "branch." This chapter formalises that restriction (the principal value branch) for all six trig functions, and then builds a toolkit of properties for simplifying and combining inverse trig expressions.

The chapter builds in a deliberate sequence: first you learn to read off the principal value of an inverse trig function directly (Exercise 2.1), then you learn the substitution technique — setting x = sinθ, cosθ or tanθ — that turns a messy algebraic expression inside an inverse trig function into a clean double- or triple-angle identity (Exercise 2.2). The Miscellaneous Exercise then combines both skills with the sum and difference formulas, including a couple of equations where checking for extraneous roots genuinely matters.

How the Chapter Builds

One Idea Leads to the Next

Restricting the Domain

Trig functions aren't one-one over ℝ, so their inverse only exists on a chosen "principal value branch."

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Principal Values

Read off the unique angle in the principal branch matching a given trig ratio. Exercise 2.1.

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Properties & Substitution

Use x = sinθ / cosθ / tanθ to collapse an expression into a double- or triple-angle identity. Exercise 2.2.

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Mixed Applications

Sum/difference formulas, harder proofs, and equations — including checking for extraneous roots. Misc. Exercise.

Quick Reference

Important Formulas — Chapter 2

Everything you need before you start solving. This is a summary for quick recall — the Formula Cards below has the full printable version for all of Relations, Functions and Trigonometry.

Principal Value Branches (§2.2)

Function	Domain	Range (principal value)
\sin^{-1}	[-1,1]	\left[-\frac{\pi}{2},\frac{\pi}{2}\right]
\cos^{-1}	[-1,1]	[0,\pi]
\text{cosec}^{-1}	\mathbb{R}-(-1,1)	\left[-\frac{\pi}{2},\frac{\pi}{2}\right]-\{0\}
\sec^{-1}	\mathbb{R}-(-1,1)	[0,\pi]-\left\{\frac{\pi}{2}\right\}
\tan^{-1}	\mathbb{R}	\left(-\frac{\pi}{2},\frac{\pi}{2}\right)
\cot^{-1}	\mathbb{R}	(0,\pi)

Basic Identities

Cancellation identities

\sin(\sin^{-1}x)=x,\;\; x\in[-1,1]

\sin^{-1}(\sin x)=x,\;\; x\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]

Similar pairs hold for each of the other five functions, restricted to their own principal range.

Don't confuse these

\sin^{-1}x \neq (\sin x)^{-1}

(\sin x)^{-1}=\dfrac{1}{\sin x}=\text{cosec}\,x — a completely different quantity from the inverse sine function.

Sum and Difference Formulas (Ex 2.2, Misc.)

Sum of two tan⁻¹ terms

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

If xy \gt 1, add or subtract \pi depending on the sign of x and y.

Difference of two tan⁻¹ terms

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

Used repeatedly in the Miscellaneous Exercise's equation-solving questions.

Double and Triple Angle Conversions

2 tan⁻¹x, three ways

2\tan^{-1}x=\sin^{-1}\left(\dfrac{2x}{1+x^2}\right)=\cos^{-1}\left(\dfrac{1-x^2}{1+x^2}\right)=\tan^{-1}\left(\dfrac{2x}{1-x^2}\right)

Each form has its own domain restriction — check it before applying.

Triple-angle conversions

3\sin^{-1}x=\sin^{-1}(3x-4x^3),\;\; x\in\left[-\tfrac12,\tfrac12\right]

3\cos^{-1}x=\cos^{-1}(4x^3-3x),\;\; x\in\left[\tfrac12,1\right]

Proved in Exercise 2.2 using the substitution x = sinθ / cosθ.

Decision Guide

Which Substitution Should I Use?

The single biggest skill in Exercise 2.2 — matching the shape of the expression to the right substitution before you touch any identity.

You see this in the expression	Try this substitution	Why
\sqrt{1-x^2} or \sqrt{a^2-x^2}	x = sinθ (or a sinθ)	Turns the square root into \cos\theta cleanly via 1-\sin^2\theta=\cos^2\theta.
\sqrt{1+x^2} or a denominator like a^2+x^2	x = tanθ (or a tanθ)	Matches the identity 1+\tan^2\theta=\sec^2\theta.
\dfrac{2x}{1+x^2} or \dfrac{1-x^2}{1+x^2}	x = tanθ	Collapses directly to \sin2\theta or \cos2\theta via the double-angle identities.
2x\sqrt{1-x^2}	x = sinθ	Becomes 2\sin\theta\cos\theta=\sin2\theta immediately.
An equation mixing inverse functions of two different variables, x and y	Convert both sides to tan⁻¹ form first	Lets you finish with the tan addition/subtraction formula in one step.

Avoid These

Common Mistakes to Avoid in This Chapter

Drawn from where students actually lose marks across both exercises and the Miscellaneous Exercise.

Assuming sin⁻¹(sin x) = x for any x. This only holds when x is already inside the principal range [−π/2, π/2] — outside it, you must first find the equivalent angle that does lie in that range.
Dropping the absolute value incorrectly when simplifying something like √(1−sin²θ) back to cosθ — always confirm the sign from the given domain before deciding it's cosθ and not −cosθ.
Confusing sin⁻¹x with (sin x)⁻¹. The second one is just 1/sin x = cosec x — an entirely different function.
Getting the sec⁻¹ and cosec⁻¹ ranges wrong. sec⁻¹'s range is [0, π] excluding π/2 (not a symmetric interval like tan⁻¹'s), and cosec⁻¹'s range is [−π/2, π/2] excluding 0.
Not verifying solutions after squaring or using a double-angle identity while solving an equation — extraneous roots are common here (see the Miscellaneous Exercise, where x = 1/2 passes the algebra but fails when substituted back into the original equation).
Applying tan⁻¹x + tan⁻¹y = tan⁻¹[(x+y)/(1−xy)] without checking xy < 1 — when xy > 1, the identity needs a ±π correction.
Forgetting to state why the resulting angle lies in the target principal range before cancelling an inverse function with its trig function — CBSE's marking scheme usually awards a step for this justification.

Solve Chapter-Wise

Class 12 Maths NCERT Solutions Chapter 2 — Choose an Exercise

2.1

Exercise 2.1

Principal values of inverse trigonometric functions — direct evaluation plus two MCQs · 14 questions

Solve Exercise 2.1 →

2.2

Exercise 2.2

Properties of inverse trigonometric functions — triple-angle proofs, substitution-based simplification · 9 questions

Solve Exercise 2.2 →

Miscellaneous Exercise

Sum/difference formula proofs, harder simplifications, and equation-solving with extraneous-root checks · 14 questions

Solve Miscellaneous →

📐 Keep the Formulas Handy

Every formula for Relations, Functions and Trigonometry — principal values, properties, and identities — in one printable PDF.

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Common Questions

Frequently Asked Questions

Quick answers about Chapter 2, Inverse Trigonometric Functions.

How many exercises are there in Chapter 2, Inverse Trigonometric Functions?

There are two main exercises — 2.1 and 2.2 — plus a Miscellaneous Exercise, totalling 37 questions across principal values, properties of inverse trigonometric functions, and mixed application problems.

Is this chapter important for the board exam?

Yes — it's usually tested with 1–2 short questions or as part of a longer proof, and the properties from Exercise 2.2 also show up indirectly in Chapter 5 (Continuity and Differentiability) whenever you differentiate an inverse trig expression.

What is the difference between sin⁻¹x and (sin x)⁻¹?

sin⁻¹x is the inverse sine function (arcsine) — it asks "which angle has this sine value?" (sin x)⁻¹ is the reciprocal of sin x, which equals 1/sin x, i.e. cosec x. The two are unrelated, and mixing them up is one of the most common errors students make in this chapter.

What should I revise before starting this chapter?

Be comfortable with the basic trigonometric ratios, the standard angles (0°, 30°, 45°, 60°, 90° and their radian equivalents), and the double- and triple-angle identities from Class 11 trigonometry — Exercise 2.2 leans on these constantly.

Where can I find the official NCERT textbook for this chapter?

Inverse Trigonometric Functions is Chapter 2 of the NCERT Class 12 Mathematics textbook (Part I), published by the National Council of Educational Research and Training (NCERT) and prescribed by CBSE. You can download the official textbook PDF directly from ncert.nic.in, NCERT's official website — the solutions on this page follow the exercises exactly as they appear there.