NCERT Solutions for Class 12 Maths Chapter 2 – Inverse Trigonometric Functions Exercise 2.2

Flashcard for Question 1

Quick Tip:
If asked to prove inverse identities, try substituting θ = sin⁻¹(x) and apply trigonometric identities like sin(3θ) to simplify.

Common Mistake:
Some students forget to restrict the domain of x, leading to incorrect inverse values. Always check if x lies within the valid range for the identity to hold.

Exam Insight:
This proof-style identity is commonly asked for 2 or 3 marks in board exams.

Flashcard for Question 2

Quick Tip:
For inverse identities, try setting θ = cos⁻¹(x) and simplify using standard trigonometric identities.

Common Mistake:
Ignoring the domain can lead to undefined or incorrect inverse values. Ensure x ∈ [1/2, 1] to keep θ within principal range of cos⁻¹.

Exam Insight:
Proofs involving inverse functions and triple angle identities are popular in CBSE long answer sections.

Flashcard for Question 3

Quick Tip:
For expressions involving √(1 + x²), try substituting x = tan θ to simplify using trigonometric identities.

Common Mistake:
Students often try to rationalize or expand directly—substitution simplifies this much faster.

Exam Insight:
This type of question appears in MCQs or 2-mark questions, testing your comfort with trigonometric substitutions.

Flashcard for Question 4

Quick Tip:
Recognize standard identities involving half-angle formulas:
√[(1 − cos x)/(1 + cos x)] = tan(x/2)

Common Mistake:
Students forget that inverse and original function cancel only when the angle lies within the principal value branch of inverse tan. Always check domain restrictions.

Exam Insight:
These half-angle simplification problems are frequently tested in 1- or 2-mark board questions.

Flashcard for Question 5

Quick Tip:
Use the identity: (1 − tan x) / (1 + tan x) = tan(π/4 − x) when dealing with expressions like this.

Common Mistake:
tan⁻¹(tan θ) = θ only when θ lies in the principal range (−π/2, π/2). Make sure π/4 − x lies in that range.

Exam Insight:
This kind of simplification is common in short answer questions on the CBSE and is also helpful in the objective parts of entrance exams.

Flashcard for Question 6

Quick Tip:
When you see x over √(a² − x²), check if it’s in the form of a standard inverse identity. Recognizing this quickly saves time.

Common Mistake:
Don’t try to rationalize or manipulate manually—this is a direct identity. Also, make sure x is within (−a, a) to satisfy domain.

Exam Insight:
This identity-based conversion is a popular in both short answer theory questions and competitive MCQs.

Flashcard for Question 7

Quick Tip:
If you see a cubic form in both numerator and denominator, look for identities of tan(3θ) or tan(θ) transformations.

Common Mistake:
Some students neglect to factor or don’t see the pattern that fits the triple angle identity, which makes the simplification wrong.

Exam Insight:
You should expect to see these kinds of identity-based transformations in 3-mark board questions or advanced MCQs. They are commonly used to assess how well you comprehend inverse trig identities.

Flashcard for Question 8

Quick Tip:
Evaluate the inverse expression first, then apply the trigonometric identity. Stick to standard angles like π/6, π/4, π/3 for quick substitution.

Common Mistake:
Some students mistakenly apply double angle identities unnecessarily. This is simpler if you evaluate step-by-step using known values.

Exam Insight:
These questions will test your understanding of both standard angle values and inverse functions. Expect questions on the board that are only worth one mark.

Flashcard for Question 9

Quick Tip:
Recognize forms of sin⁻¹(2x / (1 + x²)) and cos⁻¹((1 − y²) / (1 + y²)) as double-angle identities related to tan⁻¹(x).

Common Mistake:
Not using the right inverse trig identities or ignoring domain limits like xy < 1 might lead to wrong simplification.

Exam Insight:
This type of multi-step simplification appears in 3- or 4-mark questions and tests layered understanding of inverse trig identities and their compositions.

Flashcard for Question 10

Quick Tip:
When applying sin⁻¹(sin θ), ensure θ is in the principal range [−π/2, π/2]; otherwise, adjust using the reference angle.

Common Mistake:
Errors happen when you give the original angle as the solution without checking the principal value range.

Exam Insight:
This is a concept check on the definition of inverse functions and is common in 1-mark board or MCQ questions.

Flashcard for Question 11

Short Formula Recall:
tan⁻¹(tan θ) = θ only if θ ∈ (−π/2, π/2)

Quick Tip:
3π/4 ∉ (−π/2, π/2), so use equivalent angle in that range: tan⁻¹(tan 3π/4) = −π/4

Common Mistake:
Answering 3π/4 directly without adjusting for the principal value range of tan⁻¹.

Exam Insight:
Standard inverse check — often appears in 1-mark MCQs or concept checks.

Flashcard for Question 12

Short Formula Recall:
tan(A + B) = (tan A + tan B) / (1 − tan A·tan B)

Quick Tip:
Convert each inverse function into a triangle and find tan values before applying the tan(A + B) identity.

Common Mistake:
Forgetting to build right triangles or misapplying tan(A + B) formula.

Exam Insight:
Frequently used to test inverse-to-trig conversions and compound angle identities – common in 3-mark questions.