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CBSE · Class 11 · Mathematics

Sum and Difference of Angles (Ch 3: 3.4, part 1)

4 concepts · 32 practice questions with answers · Free to read

cos (x + y) and cos (x − y)

From daily life

Two turns added together: a quarter-turn then another small turn. The cosine of the total is not just cos x + cos y.

Simple definition

The basic results (proved with the unit circle) are: cos(x + y) = cos x cos y − sin x sin y and cos(x − y) = cos x cos y + sin x sin y. Also cos(−x) = cos x, sin(−x) = −sin x, cos(π/2 − x) = sin x and sin(π/2 − x) = cos x.

Formula: cos(x + y) = cos x cos y − sin x sin y

Rule 1: Expand — Write the formula, put in the known values, and simplify.

  • Easy: cos 75° = cos(45° + 30°) = (1/√2)(√3/2) − (1/√2)(1/2) = (√3 − 1)/(2√2) ≈ 0.2588.
  • Medium: cos 15° = cos(45° − 30°) = (√3 + 1)/(2√2) ≈ 0.9659.
  • Exam-type: cos 90° = cos(60° + 30°) = (1/2)(√3/2) − (√3/2)(1/2) = 0.

Rule 2: Compress — Spot the pattern cos A cos B ∓ sin A sin B.

  • Easy: cos 40° cos 20° − sin 40° sin 20° = cos 60° = 1/2.
  • Medium: cos 50° cos 5° + sin 50° sin 5° = cos 45° = 1/√2.
  • Exam-type: cos x cos 2x + sin x sin 2x = cos x.
Remember

The sign on the right is opposite to the sign inside the bracket for cosine.

Common mistake

Writing cos(x + y) = cos x + cos y.

Memory clue

COS opposes: + inside gives − outside.

Practice questions

Easy
  1. 1Write cos(A + B) in full.
  2. 2Find cos 15° ≈ using the formula (decimal, 4 places).
  3. 3Simplify cos 70° cos 20° − sin 70° sin 20°.
Medium
  1. 4Simplify cos 80° cos 20° + sin 80° sin 20°.
  2. 5Find cos 105° using 60° + 45°.
  3. 6Prove: cos(π/2 − x) = sin x.
Challenging
  1. 7Simplify cos(x + y) cos y + sin(x + y) sin y.
  2. 8If cos A = 3/5, cos B = 5/13 (both acute), find cos(A + B).
Show answers (try them yourself first)
  1. 1cos A cos B − sin A sin B
  2. 20.9659
  3. 3cos 90° = 0
  4. 4cos 60° = 1/2
  5. 5(1/2)(1/√2) − (√3/2)(1/√2) = (1 − √3)/(2√2) ≈ -0.2588
  6. 6cos(π/2)cos x + sin(π/2) sin x = 0 + sin x
  7. 7cos((x + y) − y) = cos x
  8. 8sin A = 4/5, sin B = 12/13: (3/5)(5/13) − (4/5)(12/13) = (15 − 48)/65 = −33/65

sin (x + y) and sin (x − y)

From daily life

Adding two turns again — the sine of the total mixes a sine and a cosine of each.

Simple definition

Using cos(π/2 − x) = sin x one gets: sin(x + y) = sin x cos y + cos x sin y and sin(x − y) = sin x cos y − cos x sin y.

Formula: sin(x + y) = sin x cos y + cos x sin y

Rule 1: Expand — Expand, then use standard values.

  • Easy: sin 75° = sin(45° + 30°) = (√3 + 1)/(2√2) ≈ 0.9659.
  • Medium: sin 15° = sin(45° − 30°) = (√3 − 1)/(2√2) ≈ 0.2588.
  • Exam-type: sin 120° = sin(90° + 30°) = cos 30° = √3/2.

Rule 2: Compress — Spot sin A cos B ± cos A sin B.

  • Easy: sin 50° cos 10° + cos 50° sin 10° = sin 60° = √3/2.
  • Medium: sin 80° cos 35° − cos 80° sin 35° = sin 45° = 1/√2.
  • Exam-type: sin 2x cos x − cos 2x sin x = sin x.
Remember

For sine the sign stays the same as in the bracket.

Common mistake

Mixing sin x sin y with sin x cos y.

Memory clue

SIN is a Sociable: Same sign, mixed sin-cos.

Spot the difference

cos(x ± y)sin(x ± y)
cos cos ∓ sin sinsin cos ± cos sin
sign flipssign stays

Practice questions

Easy
  1. 1Write sin(A − B) in full.
  2. 2Find sin 75° (4 decimal places).
  3. 3Simplify sin 25° cos 20° + cos 25° sin 20°.
Medium
  1. 4Find sin 105° using 60° + 45°.
  2. 5If sin A = 3/5, sin B = 5/13 (both acute), find sin(A + B).
  3. 6Prove: sin(π − x) = sin x.
Challenging
  1. 7Simplify sin(x + y) + sin(x − y).
  2. 8Prove: sin(x + y) sin(x − y) = sin²x − sin²y.
Show answers (try them yourself first)
  1. 1sin A cos B − cos A sin B
  2. 20.9659
  3. 3sin 45° = 1/√2
  4. 4(√3/2)(1/√2) + (1/2)(1/√2) = (√3 + 1)/(2√2) ≈ 0.9659
  5. 5cos A = 4/5, cos B = 12/13: (3/5)(12/13) + (4/5)(5/13) = (36 + 20)/65 = 56/65
  6. 6sin π cos x − cos π sin x = 0 + sin x
  7. 72 sin x cos y
  8. 8(sin x cos y)² − (cos x sin y)² = sin²x(1 − sin²y) − (1 − sin²x) sin²y = sin²x − sin²y

tan (x + y) and tan (x − y)

From daily life

The slope of a combined turn: we divide sine by cosine of the sum.

Simple definition

tan(x + y) = (tan x + tan y)/(1 − tan x tan y) and tan(x − y) = (tan x − tan y)/(1 + tan x tan y), whenever the angles and the denominators are defined.

Formula: tan(x + y) = (tan x + tan y)/(1 − tan x tan y)

Rule 1: Find a tangent — Put the known tangents into the formula.

  • Easy: tan 75° = tan(45° + 30°) = (1 + 1/√3)/(1 − 1/√3) = 2 + √3 ≈ 3.7321.
  • Medium: tan 15° = tan(45° − 30°) = 2 − √3 ≈ 0.2679.
  • Exam-type: tan(A + B) with tan A = 1/2, tan B = 1/3 equals (5/6)/(5/6) = 1, so A + B = 45°.

Rule 2: Pattern in the denominator — If x + y = 45° then (1 + tan x)(1 + tan y) = 2.

  • Easy: tan 20° + tan 25° + tan 20° tan 25° = 1.
  • Medium: If tan A = 2, tan B = 3: tan(A + B) = 5/(1 − 6) = −1.
  • Exam-type: tan(A − B) for tan A = 4, tan B = 1/2: (4 − 1/2)/(1 + 2) = 7/6.
Remember

Top: same sign as the bracket. Bottom: opposite sign.

Common mistake

Forgetting that the denominator has 1 ∓ tan x tan y.

Memory clue

tan: Top follows, Bottom flips.

Practice questions

Easy
  1. 1Write tan(A + B) in full.
  2. 2Find tan 105° using 60° + 45°.
  3. 3If tan A = 1/3 and tan B = 1/2, find tan(A + B).
Medium
  1. 4If tan A = 3 and tan B = 1/2, find tan(A − B).
  2. 5Simplify (tan 50° − tan 20°)/(1 + tan 50° tan 20°).
  3. 6Simplify (tan 35° + tan 10°)/(1 − tan 35° tan 10°).
Challenging
  1. 7If tan x = 1/7 and tan y = 1/3, find tan(x + y).
  2. 8Prove: tan 15° + tan 30° + tan 15° tan 30° = 1.
Show answers (try them yourself first)
  1. 1(tan A + tan B)/(1 − tan A tan B)
  2. 2(√3 + 1)/(1 − √3) = −(2 + √3) ≈ -3.7321
  3. 3(5/6)/(1 − 1/6) = 1
  4. 4(5/2)/(1 + 3/2) = 1
  5. 5tan 30° = 1/√3
  6. 6tan 45° = 1
  7. 7(10/21)/(1 − 1/21) = 1/2
  8. 8tan(15° + 30°) = 1, so (tan 15° + tan 30°) = 1 − tan 15° tan 30°

Using the formulas smartly

From daily life

If you know the sine and cosine of two small angles, you can build the answer for a bigger one without a calculator.

Simple definition

Typical uses: values for 15°, 75°, 105°; reduction formulas; proving identities by writing a bigger angle as a sum such as 3x = 2x + x.

Rule 1: Reduction formulas — Shift by π/2 or π.

  • Easy: sin(π/2 + x) = cos x.
  • Medium: cos(π − x) = −cos x.
  • Exam-type: tan(π + x) = tan x.

Rule 2: Prove a result — Break the angle into a sum and expand.

  • Easy: sin 3x = sin(2x + x).
  • Medium: cos 2x = cos(x + x) = cos²x − sin²x.
  • Exam-type: tan(π/4 + x) = (1 + tan x)/(1 − tan x).
Remember

Always check by putting a simple value such as x = 0 or x = 45° in both sides.

Common mistake

Assuming sin(A + B) = sin A + sin B.

Memory clue

Break the big angle, then bridge with the formula.

Practice questions

Easy
  1. 1Find sin(π/2 + 30°).
  2. 2Find cos(π − 60°).
  3. 3Prove: tan(π/4 + x) = (1 + tan x)/(1 − tan x).
Medium
  1. 4Find sin 15° cos 15°.
  2. 5Prove: cos(x + y) cos(x − y) = cos²x − sin²y.
  3. 6Find sin 15° + cos 15° (4 decimal places).
Challenging
  1. 7Prove: sin(A + B)/(cos A cos B) = tan A + tan B.
  2. 8Find the exact value of cos 15° − sin 15°.
Show answers (try them yourself first)
  1. 1cos 30° = √3/2
  2. 2−cos 60° = −1/2
  3. 3(tan π/4 + tan x)/(1 − tan π/4 tan x) with tan π/4 = 1
  4. 4(1/2) sin 30° = 1/4
  5. 5(cos x cos y)² − (sin x sin y)² = cos²x(1 − sin²y) − (1 − cos²x) sin²y = cos²x − sin²y
  6. 61.2247 (this equals √6/2)
  7. 7Expand: (sin A cos B + cos A sin B)/(cos A cos B)
  8. 8(√6+√2)/4 − (√6−√2)/4 = √2/2 ≈ 0.7071