3.4 Trigonometric Functions of Sum and Difference of Two Angles – Class 11 Mathematics

Introduction

Trigonometry is one of the most important areas of mathematics in Class 11 and Class 12, especially for students preparing for competitive exams like JEE, KCET, and COMEDK. Within NCERT Class 11 Mathematics Chapter 3, the section 3.4: Trigonometric Functions of Sum and Difference of Two Angles introduces some of the most frequently used identities in problem-solving.

The ability to simplify expressions such as sin(x+y), cos(x-y), or tan(x+y) allows us to break down complex trigonometric problems into simpler forms. These formulas not only have theoretical importance but also serve as the foundation for advanced concepts like multiple-angle formulas, transformations, and solving trigonometric equations.

For JEE aspirants, these identities are crucial for solving objective-type questions quickly and accurately. For KCET/COMEDK, where speed and accuracy play a big role, mastering these formulas gives a huge advantage. This topic builds directly from the NCERT framework, ensuring conceptual clarity that is essential for both board exams and competitive exams.

Set-up and Notation

Consider the unit circle centered at the origin.
Let the point P(cos x, sin x) correspond to angle x, and Q(cos y, sin y) correspond to angle y.
The chord (or arc) subtending the angle difference between P and Q has central angle |x-y|.
We’ll compute the squared distance PQ² in two ways and equate them to obtain the key identity for cos(x-y), then deduce the rest.

Lemma: Chord length on the unit circle
For a central angle θ, the chord length on the unit circle is 2 sin(θ/2). Hence the squared chord length is 4 sin²(θ/2). Using the identity 2 sin²(θ/2) = 1 − cos θ, we get:
4 sin²(θ/2) = 2(1 − cos θ).

So for the chord PQ subtending angle |x−y|, we have:
PQ² = 2(1 − cos(x−y)).

Derivation of cos(x−y)

Method: Compute PQ² from coordinates and from the chord formula, then equate.

Coordinate distance formula:
PQ² = (cos x − cos y)² + (sin x − sin y)²

Expanding,
PQ² = (cos²x + cos²y − 2 cos x cos y) + (sin²x + sin²y − 2 sin x sin y)

Group terms using cos²x + sin²x = 1 and cos²y + sin²y = 1:
PQ² = (1 + 1) − 2(cos x cos y + sin x sin y)
PQ² = 2 − 2(cos x cos y + sin x sin y)

Chord formula on the unit circle:
PQ² = 2(1 − cos(x−y))

Equating both:
2 − 2(cos x cos y + sin x sin y) = 2(1 − cos(x−y))

Divide by 2:
1 − (cos x cos y + sin x sin y) = 1 − cos(x−y)

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

Derivation of cos(x+y)

Use cos(−y) = cos y and sin(−y) = −sin y.

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

Derivation of sin(x+y)

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

Replacing y with −y:
sin(x−y) = sin x cos y − cos x sin y

Derivation of tan(x±y)

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

Cotangent Forms

cot(x+y) = (cot x cot y − 1) / (cot y + cot x)
cot(x−y) = (cot x cot y + 1) / (cot y − cot x)

Reciprocal Forms

sec(x±y) = 1 / cos(x±y)
csc(x±y) = 1 / sin(x±y)

Final List of Identities

• sin(x+y) = sin x cos y + cos x sin y
• sin(x−y) = sin x cos y − cos x sin y
• cos(x+y) = cos x cos y − sin x sin y
• cos(x−y) = cos x cos y + sin x sin y
• tan(x+y) = (tan x + tan y) / (1 − tan x tan y)
• tan(x−y) = (tan x − tan y) / (1 + tan x tan y)
• cot(x+y) = (cot x cot y − 1) / (cot y + cot x)
• cot(x−y) = (cot x cot y + 1) / (cot y − cot x)

Quick Reference Table

These formulas must be memorized thoroughly and practiced regularly.

Deep-Dive Examples

Example 1: Exact evaluation of sin 75°
Write 75° = 45° + 30°.
sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°

Using values:
sin 45° = √2/2, cos 45° = √2/2, cos 30° = √3/2, sin 30° = 1/2

sin 75° = (√2/2)(√3/2) + (√2/2)(1/2)
= √6/4 + √2/4
= (√6 + √2)/4

Example 2: Exact evaluation of cos 15°
Write 15° = 45° − 30°.
cos 15° = cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30°

Substitute values:
cos 15° = (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2)/4

Example 3: Prove an identity
Show that sin(A+B) sin(A−B) = sin²A − sin²B

Start with LHS:
sin(A+B) sin(A−B) = (sin A cos B + cos A sin B)(sin A cos B − cos A sin B)

= (sin A cos B)² − (cos A sin B)²
= sin²A cos²B − cos²A sin²B

Now write cos²B = 1 − sin²B and cos²A = 1 − sin²A:
= sin²A(1 − sin²B) − (1 − sin²A) sin²B
= sin²A − sin²A sin²B − sin²B + sin²A sin²B
= sin²A − sin²B

Hence proved.

Example 4: Tangent with numbers
If tan A = 2 and tan B = 3, find tan(A+B).

tan(A+B) = (tan A + tan B) / (1 − tan A tan B)
= (2 + 3) / (1 − 6)
= 5 / (−5)
= −1

Example 5: Transform to a single function
Simplify (sin x + sin y) / (cos x + cos y).

Use sum-to-product:
sin x + sin y = 2 sin((x+y)/2) cos((x−y)/2)
cos x + cos y = 2 cos((x+y)/2) cos((x−y)/2)

So,
(sin x + sin y) / (cos x + cos y) = [2 sin((x+y)/2) cos((x−y)/2)] / [2 cos((x+y)/2) cos((x−y)/2)]

= tan((x+y)/2), provided denominators ≠ 0.

Example 6: Prove product-to-sum relation
Prove cos x cos y = [cos(x+y) + cos(x−y)] / 2.

From formulas:
cos(x+y) = cos x cos y − sin x sin y
cos(x−y) = cos x cos y + sin x sin y

Add both:
cos(x+y) + cos(x−y) = 2 cos x cos y

Therefore, cos x cos y = (cos(x+y) + cos(x−y)) / 2

Example 7: Solve equation
Solve sin(x+y) = sin(x−y).

Case 1: x+y = x−y + 2kπ ⇒ 2y = 2kπ ⇒ y = kπ

Case 2: x+y = π − (x−y) + 2kπ ⇒ 2x + y = π + 2kπ ⇒ 2x + y = (2k+1)π

So solutions are y = kπ or 2x+y = (2k+1)π.

Example 8: KCET/COMEDK fast calculation
Find tan 75°.

75° = 45° + 30°
tan 75° = (tan 45° + tan 30°) / (1 − tan 45° tan 30°)

= (1 + 1/√3) / (1 − 1/√3)
= (√3+1)/√3 ÷ (√3−1)/√3
= (√3+1)/(√3−1)

Multiply by (√3+1)/(√3+1):
= (√3+1)² / (3−1) = (3 + 2√3 + 1)/2 = (4 + 2√3)/2 = 2 + √3

Example 9: JEE Main style
If sin θ = 3/5 and cos φ = 12/13 with θ, φ in (0, π/2), find cos(θ+φ).

cos θ = 4/5 (since quadrant I), sin φ = 5/13

cos(θ+φ) = cos θ cos φ − sin θ sin φ
= (4/5)(12/13) − (3/5)(5/13)
= (48 − 15)/65 = 33/65

Example 10: From compound to double angle
Simplify sin(x+y) + sin(x−y).

= (sin x cos y + cos x sin y) + (sin x cos y − cos x sin y)
= 2 sin x cos y

If y=x, then sin(2x) = 2 sin x cos x

Practice Problems

1. Evaluate sin 105°

    

2. Show sin x sin y = [cos(x−y) − cos(x+y)] / 2

    

3. If sin α = 5/13 and cos β = 3/5, find sin(α+β)

    

4. Simplify (cos x − cos y) / (sin x − sin y)

    

5. If tan θ = 4/3 and tan φ = 1/2, find tan(θ−φ)

Answers:

1. sin 105° = (√6 + √2)/4

    

2. From identities, cos(x−y) − cos(x+y) = 2 sin x sin y ⇒ proven

    

3. sin(α+β) = (5/13)(3/5) + (12/13)(4/5) = 51/65

    

4. = −tan((x+y)/2)

    

5. tan(θ−φ) = (4/3 − 1/2) / (1 + (4/3)(1/2)) = (5/6)/(10/6) = ½

Applications to JEE / KCET / COMEDK

• Direct value evaluations:
  Compute exact values like sin 75°, cos 15°, tan 75°, sin 105° by rewriting them as sums/differences of standard angles (30°, 45°, 60°, 90°).

   

• Fast simplification:
  Spot patterns quickly:
  • sin x cos y + cos x sin y → sin(x + y)
  • cos x cos y − sin x sin y → cos(x + y)
  • sin x cos y − cos x sin y → sin(x − y)
  • cos x cos y + sin x sin y → cos(x − y)

   

• Products ↔ sums:
  Use product-to-sum when helpful:
  • cos x cos y = [cos(x + y) + cos(x − y)] / 2
  • sin x sin y = [cos(x − y) − cos(x + y)] / 2

   

• Build to double/multiple angles:
  Put y = x to get double-angle formulas (e.g., sin 2x = 2 sin x cos x; cos 2x = cos²x − sin²x). Repeated use leads to triple/multiple-angle relations.

   

• Be careful with tangent forms:
  tan(x ± y) = (tan x ± tan y) / (1 ∓ tan x tan y).
  Always check the denominator first.

   

• Quick checks to verify work:
  Substitute y = 0 or x = y to see if an identity collapses to a known truth.
  

Common Pitfalls and How to Avoid Them

1. Sign confusion
   • sine “stays same”: sin(x + y) = sin x cos y + cos x sin y
   • cosine “changes sign”: cos(x + y) = cos x cos y − sin x sin y
   Tip: write the two side-by-side in your formula sheet to imprint the pattern.

    

2. Tangent denominator errors
   • tan(x + y): denominator is 1 − tan x tan y
   • tan(x − y): denominator is 1 + tan x tan y
   A flipped sign is a common source of mistakes.

    

3. Ignoring domain/restrictions
   Expressions like tan(x + y) or sin(x + y)/cos(x + y) are undefined when the denominator is 0. In MCQs, this often helps eliminate wrong options.

    

4. Wrong quadrant for exact values
   When evaluating angles like 105° or −15°, confirm the quadrant so you get the correct sign.

    

5. Over-expanding instead of compressing
   Many expressions simplify instantly if you recognize the compound-angle pattern. Look first for sin x cos y ± cos x sin y and cos x cos y ∓ sin x sin y structures.
   

Strategy Checklist for Exams

• Rewrite non-standard angles:
  75° = 45° + 30°, 105° = 60° + 45°, 15° = 45° − 30°.
• Compress before you expand:
  Convert sin x cos y ± cos x sin y to sin(x ± y) and cos x cos y ∓ sin x sin y to cos(x ± y).
• Use product-to-sum when needed:
  Helpful for simplifying products or matching answer forms.
• Control tangent carefully:
  Before using tan(x ± y), check the denominator (1 ∓ tan x tan y). If risky, switch all terms to sine/cosine first.
• Leverage right-triangle thinking:
  If sin θ = a/b or cos θ = a/b is given, draw a quick reference triangle to find the missing side and the complementary ratio.
• Do a sanity check:
  Plug in y = 0 or x = y to see if your result matches a known identity (like sin 2x or cos 2x).

Summary and Key Takeaways

• Core identities:
  • sin(x ± y) = sin x cos y ± cos x sin y
  • cos(x ± y) = cos x cos y ∓ sin x sin y
  • tan(x ± y) = (tan x ± tan y) / (1 ∓ tan x tan y)
• Derivation logic (NCERT-aligned):
  Compare the squared chord length on the unit circle with the coordinate distance formula to obtain cos(x − y); derive the rest by symmetry and co-function ideas.
• Bridges to other topics:
  These formulas directly yield double-angle, triple-angle, and product-to-sum identities that appear frequently in JEE, KCET, and COMEDK.
• Accuracy boosters:
  Memorize sign patterns, watch denominators, and always consider quadrant signs for exact angles.
• Speed tips:
  Pattern-spotting (compressing to sin(x ± y), cos(x ± y)) and quick checks (y = 0, x = y) save time and prevent errors.