Exercise 10.4 – Product to Sum and Sum to Product Formulas | First Year Mathematics 2025

Introduction

Trigonometry has many powerful formulas that allow us to change the form of expressions for easier calculation. In the New First Year Math 2025 Edition, Exercise 10.4 focuses on expressing the product of sines and cosines as a sum or difference. This is done using special formulas called Product-to-Sum formulas.

These formulas are very useful in:

Simplifying trigonometric expressions

Solving trigonometric equations

Converting complex products into simple addition/subtraction forms

Integration and physics problems in higher studies

This exercise is very important for short and long questions in board exams. Many MCQs and past paper problems are also based on these formulas. The examples in the book are especially important for understanding how to apply these formulas correctly.

Product-to-Sum Formulas

The idea is to change the product of two trigonometric functions (sin × sin, cos × cos, or sin × cos) into a sum or difference of sines or cosines.

1. Product of Sines

sin A × sin B = 1/2 [cos(A – B) – cos(A + B)]

2. Product of Cosines

cos A × cos B = 1/2 [cos(A – B) + cos(A + B)]

3. Product of Sine and Cosine

sin A × cos B = 1/2 [sin(A + B) + sin(A – B)]

cos A × sin B = 1/2 [sin(A + B) – sin(A – B)]

Sum-to-Product Formulas (Reverse Process)

In some questions, you might need to go in the reverse direction — from sums/differences to products.

1. Difference of Cosines

cos A – cos B = –2 sin((A + B)/2) × sin((A – B)/2)

2. Sum of Cosines

cos A + cos B = 2 cos((A + B)/2) × cos((A – B)/2)

3. Sum of Sines

sin A + sin B = 2 sin((A + B)/2) × cos((A – B)/2)

4. Difference of Sines

sin A – sin B = 2 cos((A + B)/2) × sin((A – B)/2)

Why These Formulas Are Important

Board Exams – Questions often ask you to prove identities or simplify using these formulas.

Physics Applications – Wave equations often use product-to-sum identities.

Higher Mathematics – In integration and Fourier analysis, these formulas make calculations easier.

Past Paper Repetition – Many repeated questions in both short and long format.

Step-by-Step Approach to Solve

1. Identify the type of product – Check if it is sin × sin, cos × cos, or sin × cos.

2. Select the right formula – Use the correct product-to-sum formula.

3. Apply angle addition and subtraction – Calculate A + B and A – B.

4. Write in sum/difference form – Replace the product with the formula result.

5. Simplify if needed – Reduce fractions or use basic identities like sin²θ + cos²θ = 1.

Example 1 – Product of Sines

Question: Express sin 40° × sin 20° as a sum or difference.

Solution:

Formula: sin A × sin B = 1/2 [cos(A – B) – cos(A + B)]

Here A = 40°, B = 20°

= 1/2 [cos(40° – 20°) – cos(40° + 20°)]

= 1/2 [cos 20° – cos 60°]

Answer: (1/2)(cos 20° – cos 60°)

Example 2 – Product of Cosines

Question: Express cos 50° × cos 10° as a sum.

Solution:

Formula: cos A × cos B = 1/2 [cos(A – B) + cos(A + B)]

Here A = 50°, B = 10°

= 1/2 [cos(50° – 10°) + cos(50° + 10°)]

= 1/2 [cos 40° + cos 60°]

Example 3 – Product of Sine and Cosine

Question: Express sin 70° × cos 10° as a sum.

Solution:

Formula: sin A × cos B = 1/2 [sin(A + B) + sin(A – B)]

Here A = 70°, B = 10°

= 1/2 [sin(70° + 10°) + sin(70° – 10°)]

= 1/2 [sin 80° + sin 60°]

Example 4 – Cos × Sin Form

Question: Express cos 40° × sin 10° as a difference.

Solution:

Formula: cos A × sin B = 1/2 [sin(A + B) – sin(A – B)]

Here A = 40°, B = 10°

= 1/2 [sin(40° + 10°) – sin(40° – 10°)]

= 1/2 [sin 50° – sin 30°]

How to Remember the Formulas

For sin × sin → cos – cos (minus between them)

For cos × cos → cos + cos (plus between them)

For sin × cos → sin + sin

For cos × sin → sin – sin

PDF SOLUTION:

Also remember: the angle differences and sums come inside cos or sin functions, and everything has a factor of 1/2.

Common Mistakes to Avoid

1. Mixing up formulas – Don’t use sin × sin formula for cos × cos questions.

2. Wrong order in subtraction – cos(A – B) is not the same as cos(B – A) in some cases.

3. Forgetting 1/2 factor – Always include 1/2 in the answer.

4. Wrong quadrant thinking – If angles go above 90°, be careful with sine and cosine signs.

Tips for Board Exam Preparation

Memorize all product-to-sum and sum-to-product formulas daily.

Practice all textbook examples — they often appear in exams.

Solve past paper problems for speed and accuracy.

Always write each step clearly in proofs to avoid losing marks.

Conclusion

Exercise 10.4 in the New First Year Math 2025 Edition is a formula-based exercise that teaches you how to convert products into sums/differences and vice versa. This skill is essential for board exam success and is also widely used in physics and higher mathematics.

By mastering these formulas and practicing regularly, you will be able to:

Solve identity proofs quickly.

Simplify trigonometric expressions easily.

Handle both short and long questions with confidence.

Remember — examples in this exercise are as important as the questions, so practice them thoroughly.