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Identities and Properties of Inverse Trigonometric Functions

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

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

    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.

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

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

Definition

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

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.

Rule to remember

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.

Memory hook

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

Examples and method

Worked example

Simplify cos⁻¹x − sin⁻¹x for x ∈ [−1,1]. Since sin⁻¹x + cos⁻¹x = π/2, cos⁻¹x = π/2 − sin⁻¹x. Therefore cos⁻¹x − sin⁻¹x = π/2 − 2sin⁻¹x. The simplified form is π/2 − 2sin⁻¹x.

Method to apply

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.

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

Find tan⁻¹(√3) + cot⁻¹(√3).

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

Answer writing and exam use

1-mark answer

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

2-mark answer

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

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. 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. Simplify cos⁻¹x − sin⁻¹x for x ∈ [−1,1]. Since sin⁻¹x + cos⁻¹x = π/2, cos⁻¹x = π/2 − sin⁻¹x. Therefore cos⁻¹x − sin⁻¹x = π/2 − 2sin⁻¹x. The simplified form is π/2 − 2sin⁻¹x. Questions ask students to simplify expressions, prove identities, convert between inverse functions, or evaluate exact values using complementary relations. Writing cos⁻¹x = sin⁻¹x − π/2 instead of cos⁻¹x = π/2 − sin⁻¹x.
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