Product to Sum Formulas With Solved Examples

Exams Prep Master

The product to sum formulas are derived from sum and difference trigonometry formulas. Henceforth, they are a set of formulas also known as trigonometric identities. These formulas are some of the most useful formulas in trigonometric functions. Let us learn more about the Product to Sum formula along with the solved questions and overall concept.

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Table of Content

What is the use of the Product to Sum formula?
Product to Sum Formula
How Product to Sum formulas are derived (derivation of formula)
Things to Remember
Sample Questions

Key takeaways: Product to Sum formula, Trigonometric identities, products of Sin and Cos, Critical trigonometry function, trigonometry functions.

Use of the Product to Sum Formula?

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The product to sum formulas is used to rewrite the product of sine and cosine functions as a sum. Sine and Cosine being some often operated trigonometric functions, it becomes necessary to know the Product to Sum formula. This formula is used to express the products as Sums. That is the basic reason behind the name of this formula i.e., Product to Sum formula.

One can also simplify the critical trigonometry function by using this formula. So, all one needs to know is the formula and its correct application. Now, have a look at the unique derivation of the Product to Sum formula.
Also read:

Signum Function

CoPlanar Vectors

Geometric Mean

Product to Sum Formula

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There are four formulas to express a product as a sum. These Product to Sum formulas are widely used as trigonometric identities.

First Formula:

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

Second Formula:

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

Third Formula:

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

Fourth Formula:

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

Also Check:

Square Root	Degree of Polynomials
Substitution Method	Cross Multiplication

How are Product to Sum formulas Derived?

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To understand the derivation of Product to Sum formulas, we need to first recall the sum and difference formulas of Sin and Cos. The sum and difference formulas in trigonometry are:

sin (A + B) = sin A cos B + cos A sin B (1)

sin (A - B) = sin A cos B - cos A sin B (2)

cos (A + B) = cos A cos B - sin A sin B (3)

cos (A - B) = cos A cos B + sin A sin B (4)

Adding or subtracting any two formulas out of these four formulas, one can easily reach the derivation of the product to sum formulas.

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Derivation of the First Formula

sin A cos B = (1/2) [ sin (A + B) + sin (A - B) ]:

Putting the equations (1) and (2) together, we have,

sin (A + B) + sin (A - B) = 2 sin A cos B

Now, if we give a divide to both the sides by 2,

sin A cos B = (1/2) [ sin (A + B) + sin (A - B) ] (First Formula)

Derivation of the Second Formula

cos A sin B = (1/2) [ sin (A + B) - sin (A - B) ]:

Subtracting equation (2) from equation (1),

sin (A + B) - sin (A - B) = 2 cos A sin B

Giving a divide to both sides by 2,

cos A sin B = (1/2) [ sin (A + B) - sin (A - B) ] (Second Formula)

Check Important Notes for Circular Representation of Inverse Trigonometric Functions

Derivation of the Third Formula

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

Putting the equations (3) and (4) together, we have

cos (A + B) + cos (A - B) = 2 cos A cos B

Now, we will divide both sides by 2,

cos A cos B = (1/2) [ cos (A + B) + cos (A - B) ] (Third Formula)

Derivation of the Fourth Formula

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

Subtracting equation (3) from equation (4),

cos (A - B) - cos (A + B) = 2 sin A sin B
Giving a divide to both sides by 2,

sin A sin B = (1/2) [ cos (A - B) - cos (A + B) ] (Fourth Formula)

Also Read More:

Graphs of Inverse Trigonometric Functions	Inverse Trigonometry
Negative of a Vector	Rolle’s Theorem

Things to Remember

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Product to sum formulas are used to write products of trigonometry functions as a sum.
There are four products to sum formulas. One can add or subtract any of them according to the problem and get the solution.
sin A cos B = (1/2) [ sin (A + B) + sin (A - B) ]
cos A sin B = (1/2) [ sin (A + B) - sin (A - B) ]
cos A cos B = (1/2) [ cos (A + B) + cos (A - B) ]
sin A sin B = (1/2) [ cos (A - B) - cos (A + B) ]
Sum and difference formulas for the derivation of product to sum formulas are:

sin (A + B) = sin A cos B + cos A sin B (1)

sin (A - B) = sin A cos B - cos A sin B (2)

cos (A + B) = cos A cos B - sin A sin B (3)

cos (A - B) = cos A cos B + sin A sin B (4)

Read More: Cubic Polynomials

Sample Questions

Ques: Find the value of sin 75° sin 15° without literally evaluating the values of sin 75° and sin 15°. (3 marks)

Ans:

Values to find: sin 75° sin 15°

We will use one of the product to sum formulas,

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

Substitute A = 75° and B = 15°, we have

sin 75° sin 15° = (1/2) [ cos (75°- 15°) - cos (75° + 15°) ]

= (1/2) [ cos 60° - cos 90°]

= (1/2) [ (1/2) - 0] (from trigonometry table)

= 1/4

The answer will be sin 75° sin 15°= 1/4.

Ques: Express Product 2 cos 5x sin 2x as a sum/difference. (3 marks)

Ans:

Already given, Product: 2 cos 5x sin 2x

We will write one of the product to sum formulas here,

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

Now, substitute A = 5x and B = 2x in the above formula,

cos 5x sin 2x = (1/2) [ sin (5x + 2x) - sin (5x - 2x) ]

cos 5x sin 2x = (1/2) [sin 7x - sin 3x]

We will multiply both the sides by 2,

2 cos 5x sin 2x = sin 7x - sin 3x

The answer will be 2 cos 5x sin 2x = sin 7x - sin 3x.

Ques: Find the value of the integral ∫ sin 3x cos 4x dx. (4 marks)

Ans:

Already given, ∫ sin 3x cos 4x dx

Using any of the product to sum formulas,

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

Exchange A as 3x and B as 4x on both sides,

sin 3x cos 4x = (1/2) [ sin (3x + 4x) + sin (3x - 4x) ] = (1/2) [ sin 7x - sin x] (because sin (-x) = - sin x).

Now, we will check the given integral using the above value.

∫ sin 3x cos 4x dx = ∫ (1/2) [ sin 7x - sin x] dx

= (1/2) [ -cos (7x) / 7 + cos x] + C (after using integration by substitution)

The answer will be sin 3x cos 4x dx = (1/2) [ -cos (7x) / 7 + cos x] + C

Ques: Simplify the function cos (3x) sin (2x) using the product to sum formula. (4 marks)

Ans:

Already Given, cos(3x) sin(2x)

Using a product to sum formula,

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

Substitute A as 3x and B as 2x on both the sides

cos 3x sin 2x = (1/2) [ sin (3x+ 2x) - sin (3x - 2x) ]

cos 3x sin 2x = (1/2) [ sin (5x) - sin (x) ]

Ques: Write cos 3 x cos 2 x as a sum. (3 marks)

Ans:

Already given: cos 3x cos 2x

Using a product to sum formula,

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

Substitute A as 3x and B as 2x on both the sides

cos 3x Cos 2x = (1/2) [ cos (3x+ 2x) - cos (3x - 2x) ]

cos 3x cos 2x = (1/2) [ cos (5x) - cos (x) ]

Therefore, our answer would be cos 3x cos 2x = (1/2) [ cos (5x) - cos (x) ]

Ques: Verify that sinx + sin y = (½) [sin (x+y) cos (x-y) ]. (3 marks)

Ans:

We have to verify: sinx + sin y = (½) [sin (x+y) cos (x-y) ].

Using a product to sum formula,

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

If x = a+b and y= a-b

Then a = ½ (x+y) and b= ½ x-y

(½ ) [sin x+y cos x-y] = ½ (sin x + sin y)

Hence, it is verified that sin x+ sin y = (½) sin+y cos x-y

Ques: Write the difference cos 8α − cos 2α as a product. (3 marks)

Ans: Sum: cos 8a - cos 2a.

For converting it as a product, we need to first use the product to sum formula of cosine…

cos a - cos b = (½) sin a+b sin a-b

Putting the values,

Therefore,

cos 8a - cos 2a = - 2 sin 5a sin 3a

Ques: Write the following product of cosines as a sum: 2cos (7x2) cos 3x 2. (3 marks)

Ans: Already given, product: (½) [cos (7x ) cos 3x]

We would be writing the formula for the product to sum of cosines now:

cos a cos b = (½) [cos(a−b )+ cos(a-b]

Now, we will substitute a and b in the formula with the values given and simplify the formula.

cos 7x cos 3x = ½ cos 7x-3x + cos 7x-3x

cos 7x cos 3x= (½) [cos 4x + cos 10x]

cos 7x cos 3x = (½) [cos 2x + cos 5 x]

Ques: Express Product 2 cos 7x sin 2x as a sum/difference. (3 marks)

Ans: Already given, Product: 2 cos 7x sin 2x

We will write one of the product to sum formulas here,

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

Now, substitute A = 7x and B = 2x in the above formula,

cos 7x sin 2x = (1/2) [ sin (7x + 2x) - sin (7x - 2x) ]

cos 7x sin 2x = (1/2) [sin 9x - sin 5x]

We will multiply both the sides by 2,

2 cos 7x sin 2x = sin 9x - sin 5x

The answer will be 2 cos 5x sin 2x = sin 9x - sin 5x.

Ques: Simplify the function cos 45° sin 15°without using multiplication. (3 marks)

Ans: Values to rewrite: cos 45° sin 15°

We will use the second formula out of the four product to sum formulas, as it has cos and sin in it just like the value given in the Question.

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

Substitute A = 45° and B = 15°, we have

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

cos 45° sin 15° =(½) [sin (45°+15°) - sin (45° - 15°)]

sin 45° sin 15° = (1/2) [ sin (60°) - sin (30°) ]

= (½) [ (√3/2) - ½]

The answer will be sin 45° sin 15° = (½) [ (√3/2) - ½ ]

Mathematics Related links:

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Factorial formula	Inverse Matrix formula	Bayes theorem formula

Product to Sum Formulas With Solved Examples

Use of the Product to Sum Formula?

Product to Sum Formula

How are Product to Sum formulas Derived?

Derivation of the First Formula

Derivation of the Second Formula

Derivation of the Third Formula

Derivation of the Fourth Formula

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

2.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

3.
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

4.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

5.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

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Product to Sum Formulas With Solved Examples

Use of the Product to Sum Formula?

Product to Sum Formula

How are Product to Sum formulas Derived?

Derivation of the First Formula

Derivation of the Second Formula

Derivation of the Third Formula

Derivation of the Fourth Formula

Things to Remember

Sample Questions

CBSE CLASS XII Related Questions

1.Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

2. Find: The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

3.A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

4.Evaluate:\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

5. Find: If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6. Find: If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).

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1.
Find a point on the line \( \frac{x - 2}{3} = \frac{1 - y}{2} = \frac{z - 3}{2} \) at a distance of \( \sqrt{2} \) units from the point \( (1, 2, 3) \).

2.
Find:

The shortest distance between the lines: \[ \vec{r}=(4+\lambda)\hat{i}+(2\lambda-1)\hat{j}-3\lambda\hat{k} \] and \[ \vec{r}=(1+2\mu)\hat{i}+(4\mu-1)\hat{j}+(2-5\mu)\hat{k} \]

3.
A carpenter needs to design a wooden box in the shape of a cuboid such that the sum of its length and breadth is 3 cm more than its height. Twice of its length, thrice of its breadth and its height add up to 10 cm. Its breadth added to 7 times its height is 1 cm less than 3 times its length.

4.
Evaluate:
\[ \int_{0}^{1} \frac{x \tan^{-1}x}{(1+x^2)^{3/2}}\,dx \]

5.
Find:

If \[ (3\hat{i}-2\hat{j}+5\hat{k})\times(4\hat{i}+p\hat{j}+q\hat{k})=\vec{0} \] then find the values of \(p\) and \(q\).

6.
Find:

If the function \[ f(x)= \begin{cases} \frac{\sin x}{x}+\cos x, & x\neq0\\ k, & x=0 \end{cases} \] is continuous at \(x=0\), then find the value of \(k\).