C
CraftExam
high importancemedium8 min

Sum and Difference Formulae for Inverse Trigonometric Functions

Sum and difference formulae combine two inverse trigonometric expressions into a single inverse expression, with conditions determined by the principal value branch.

Practice This Concept

Learn the concept

Student-friendly explanation

For tan⁻¹ expressions, the formula resembles the tangent addition formula, but the answer must lie in the principal range of tan⁻¹x. If the combined angle falls outside (−π/2, π/2), an adjustment by π may be needed depending on the signs and the value of xy.

How to write this in exams

  1. 1

    Start with the exact idea

    Sum and difference formulae combine two inverse trigonometric expressions into a single inverse expression, with conditions determined by the principal value branch.

  2. 2

    Then show how to use it

    Identify x and y, compute xy, choose the correct sum or difference condition, substitute into the fraction, simplify the inverse value, then adjust by π if the branch and signs require it.

  3. 3

    Add one concrete example

    tan⁻¹(1/2) + tan⁻¹(1/3) = tan⁻¹((1/2 + 1/3)/(1 − 1/6)) = tan⁻¹(1) = π/4 because xy = 1/6 < 1.

  4. 4

    Avoid this incomplete answer

    Giving tan⁻¹2 + tan⁻¹3 = −π/4 by using the fraction but ignoring that the actual sum is positive and greater than π/2.

Definition

Sum and difference formulae combine two inverse trigonometric expressions into a single inverse expression, with conditions determined by the principal value branch.

Example

tan⁻¹(1/2) + tan⁻¹(1/3) = tan⁻¹((1/2 + 1/3)/(1 − 1/6)) = tan⁻¹(1) = π/4 because xy = 1/6 < 1.

Rule to remember

For real x,y with xy < 1: tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)). For xy > 1 and x,y > 0, tan⁻¹x + tan⁻¹y = π + tan⁻¹((x+y)/(1−xy)) so that the angle lies in the correct interval for the sum. Difference formula: tan⁻¹x − tan⁻¹y = tan⁻¹((x−y)/(1+xy)) when the branch condition is satisfied. Analogous sin⁻¹ and cos⁻¹ combinations require checking domain and sign before using derived forms.

Memory hook

The tangent fraction gives a candidate angle; the branch check decides the final angle.

Examples and method

Worked example

Evaluate tan⁻¹2 + tan⁻¹3. Here x = 2, y = 3, so xy = 6 > 1 and both are positive. Compute (x+y)/(1−xy) = 5/(1−6) = −1. The direct tan⁻¹ value is tan⁻¹(−1) = −π/4, but the sum of two positive acute angles is greater than π/2. Add π: π − π/4 = 3π/4. Therefore tan⁻¹2 + tan⁻¹3 = 3π/4.

Method to apply

Identify x and y, compute xy, choose the correct sum or difference condition, substitute into the fraction, simplify the inverse value, then adjust by π if the branch and signs require it.

Diagram support

Not required for routine algebraic use. A unit circle can support branch checking when the resulting angle crosses π/2.

How CBSE asks it

Exam questions ask for exact evaluation, proof of an identity, or reduction of multiple inverse tangent terms to a single angle.

Avoid common mistakes

Common confusion

Applying tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) without checking whether xy < 1, xy = 1 or xy > 1.

Common wrong answer

Giving tan⁻¹2 + tan⁻¹3 = −π/4 by using the fraction but ignoring that the actual sum is positive and greater than π/2.

Exam tip

For tan⁻¹ sums, calculate xy before simplifying the fraction. The value of xy tells whether the direct principal-value formula is enough.

Quick check

Evaluate tan⁻¹(1) + tan⁻¹(1).

π/2. The direct fraction has denominator 1 − 1 = 0, and the two angles are π/4 + π/4 = π/2.

Answer writing and exam use

1-mark answer

Sum and difference formulae combine two inverse trigonometric expressions into a single inverse expression, with conditions determined by the principal value branch.

2-mark answer

Sum and difference formulae combine two inverse trigonometric expressions into a single inverse expression, with conditions determined by the principal value branch. For real x,y with xy < 1: tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)). For xy > 1 and x,y > 0, tan⁻¹x + tan⁻¹y = π + tan⁻¹((x+y)/(1−xy)) so that the angle lies in the correct interval for the sum. Difference formula: tan⁻¹x − tan⁻¹y = tan⁻¹((x−y)/(1+xy)) when the branch condition is satisfied. Analogous sin⁻¹ and cos⁻¹ combinations require checking domain and sign before using derived forms. tan⁻¹(1/2) + tan⁻¹(1/3) = tan⁻¹((1/2 + 1/3)/(1 − 1/6)) = tan⁻¹(1) = π/4 because xy = 1/6 < 1.

3-mark answer

For tan⁻¹ expressions, the formula resembles the tangent addition formula, but the answer must lie in the principal range of tan⁻¹x. If the combined angle falls outside (−π/2, π/2), an adjustment by π may be needed depending on the signs and the value of xy. For real x,y with xy < 1: tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)). For xy > 1 and x,y > 0, tan⁻¹x + tan⁻¹y = π + tan⁻¹((x+y)/(1−xy)) so that the angle lies in the correct interval for the sum. Difference formula: tan⁻¹x − tan⁻¹y = tan⁻¹((x−y)/(1+xy)) when the branch condition is satisfied. Analogous sin⁻¹ and cos⁻¹ combinations require checking domain and sign before using derived forms. Evaluate tan⁻¹2 + tan⁻¹3. Here x = 2, y = 3, so xy = 6 > 1 and both are positive. Compute (x+y)/(1−xy) = 5/(1−6) = −1. The direct tan⁻¹ value is tan⁻¹(−1) = −π/4, but the sum of two positive acute angles is greater than π/2. Add π: π − π/4 = 3π/4. Therefore tan⁻¹2 + tan⁻¹3 = 3π/4. Exam questions ask for exact evaluation, proof of an identity, or reduction of multiple inverse tangent terms to a single angle. Giving tan⁻¹2 + tan⁻¹3 = −π/4 by using the fraction but ignoring that the actual sum is positive and greater than π/2.
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