C
CraftExam
medium difficulty7 min

Identities and Properties of Inverse Trigonometric Functions

Inverse trigonometric identities are relations between inverse functions that hold under specified domain and branch conditions.

In one line: Complementary inverse pairs add to π/2, but only when the input and branch allow the statement.

Practice This Concept

Understand it

Complementary identities work because sine and cosine, tan and cot, sec and cosec are linked through complementary angles. However, inverse functions return principal values, so the conditions on x and the range of each function must be respected.

Example

For x ∈ [−1,1], sin⁻¹x + cos⁻¹x = π/2. If x = 1/2, then sin⁻¹(1/2) + cos⁻¹(1/2) = π/6 + π/3 = π/2.

Formula and rule

Key identities: sin⁻¹x + cos⁻¹x = π/2 for x ∈ [−1,1]; tan⁻¹x + cot⁻¹x = π/2 for x ∈ R; sec⁻¹x + cosec⁻¹x = π/2 for |x| ≥ 1, with branch conventions respected.

Conversion example: sin⁻¹x = cos⁻¹(√(1−x²)) only when x ∈ [0,1]; signs and intervals must be checked for other cases.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Simplify cos⁻¹x − sin⁻¹x for x ∈ [−1,1].

  1. 1Since sin⁻¹x + cos⁻¹x = π/2, cos⁻¹x = π/2 − sin⁻¹x.
  2. 2Therefore cos⁻¹x − sin⁻¹x = π/2 − 2sin⁻¹x.
  3. 3The simplified form is π/2 − 2sin⁻¹x.

Method you can reuse

Check the domain of the variable, choose a valid identity, replace one inverse function using the identity, simplify algebraically, and verify that the final expression respects the principal range.

Visual tip

A diagram is not essential for every identity, but a right triangle or unit-circle branch marking helps justify sign choices in conversions.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Find tan⁻¹(√3) + cot⁻¹(√3).

Think first, then show answer
π/2, because tan⁻¹x + cot⁻¹x = π/2 for real x under the standard principal branches.

Common wrong answer

Writing cos⁻¹x = sin⁻¹x − π/2 instead of cos⁻¹x = π/2 − sin⁻¹x.

Don't mix this up

Using an identity outside its domain, such as applying sin⁻¹x + cos⁻¹x when x is not in [−1,1].

Exam tip

In proof or simplification questions, state the substitution angle and its allowed interval. This shows the branch condition clearly.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Questions ask students to simplify expressions, prove identities, convert between inverse functions, or evaluate exact values using complementary relations.
  1. 1

    Enough for 1 mark

    Inverse trigonometric identities are relations between inverse functions that hold under specified domain and branch conditions.

  2. 2

    Add for 2 marks

    Example: For x ∈ [−1,1], sin⁻¹x + cos⁻¹x = π/2. If x = 1/2, then sin⁻¹(1/2) + cos⁻¹(1/2) = π/6 + π/3 = π/2.

  3. 3

    Add for 3 marks

    Complementary identities work because sine and cosine, tan and cot, sec and cosec are linked through complementary angles.

MCQ Quiz

Practice this concept with focused MCQs

Open the concept quiz intro first, review the test details, and then start a focused MCQ set from this concept only. Instant score and answer review are live now.

10 MCQs5 MinutesInstant Results
Practice This Concept

Help improve this page

Found something confusing, incorrect, or missing?

Practice This Concept