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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.

In one line: The tangent fraction gives a candidate angle; the branch check decides the final angle.

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Understand it

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.

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.

Formula and rule

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.

See it worked out

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

Worked example

Evaluate tan⁻¹2 + tan⁻¹3.

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

Method you can reuse

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.

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Evaluate tan⁻¹(1) + tan⁻¹(1).

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

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.

Don't mix this up

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

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.

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

Exam questions ask for exact evaluation, proof of an identity, or reduction of multiple inverse tangent terms to a single angle.
  1. 1

    Enough for 1 mark

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

  2. 2

    Add for 2 marks

    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.

  3. 3

    Add for 3 marks

    For tan⁻¹ expressions, the formula resembles the tangent addition formula, but the answer must lie in the principal range of tan⁻¹x.

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