Master Inverse Trigonometric Functions (Class 12 Maths Chapter 2) – Formulas, NCERT Solutions & CBSE PYQs

Chapter 2: Inverse Trigonometric Functions (Class 12 CBSE Math)

✅ A. Key Concepts & Formulas

1. Definition & Principal Values
   Inverse trig functions are the restricted inverses of trig functions:

   • $\sin^{-1} x$: domain $x \in [-1,1]$; range $[-π/2, π/2]$

   • $\cos^{-1} x$: domain $[-1,1]$; range $[0, π]$

   • $\tan^{-1} x$: domain all real; range $(-π/2, π/2)$

   • Similar definitions apply for $\cot^{-1}, \sec^{-1}, \csc^{-1}$ .

2. Properties & Identities

   • Odd–even behavior: $\sin^{-1}(-x)=-\sin^{-1}x$, $\cos^{-1}(-x)=π-\cos^{-1}x$, etc.

   • Complementary relations: $\sin^{-1}x + \cos^{-1}x = π/2$; $\tan^{-1}x + \cot^{-1}x = π/2$.

   • Reciprocal links: $\sin^{-1}(1/a) = \csc^{-1}(a)$ for $|a|≥1$, etc.

3. Composite Expressions
   Use algebraic identities and angle-sum formulas to simplify complex arctrig expressions:

   • $2\tan^{-1}x = \tan^{-1}\left(\frac{2x}{1 - x^2}\right)$ (for $|x|<1$)

   • $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1 - xy}\right)$ (if $xy<1$) .

4. Graphs

   • $\sin^{-1}x$: increasing from $[-1,1]$ to $[-π/2, π/2]$.

   • $\cos^{-1}x$: decreasing from $[-1,1]$ to $[π, 0]$ .

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✅ B. NCERT Exercises – Quick Overview

• Ex 2.1: Domain, range, principal values.

• Ex 2.2: Prove identities of form $\sin^{-1}(\sin x) = ...$.

• Ex 2.3: Simplify algebraic combinations of inverse trigs.

• Ex 2.4: Solve equations involving inverse trigs.

Detailed solutions are available via Teachoo and MathonGo.

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✅ C. CBSE PYQs + Best-Scoring Responses

Below are top-scoring answers to major board questions:

🔸 PYQ 2023

Q: $\cos^{-1}(\sqrt{3}/2)+\cos^{-1}(-1/2)$
A: Use principal values: $\cos^{-1}(\sqrt{3}/2)=π/6$, $\cos^{-1}(-1/2)=2π/3$ → Sum = $\boxed{5π/6}$ .

🔸 PYQ 2021

Q: $\tan^{-1}(1)+\tan^{-1}(2)+\tan^{-1}(3)= ?$
A: Combine stepwise using sum formulas → result = $\boxed{π}$ .

🔸 PYQ 2020

Q: Solve $2\tan^{-1}(\cos x) = \tan^{-1}(2\csc x)$.
A: Use relation $2\tan^{-1}y = \tan^{-1}(\frac{2y}{1-y^2})$, substitute $\cos x$, simplify gives solutions $x=0,π/4$ .

🔸 PYQ 2019

Q: Prove $\tan^{-1}(1/5)+\tan^{-1}(1/7)+\tan^{-1}(1/3)+\tan^{-1}(1/8)=π/4$.
A: Group and apply addition formulas in stages using arctan sum rules .

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