Sum to Product Formula
The sum to product formula in trigonometry are formulas that are used to express the sum and difference of sines and cosines as products of sine and cosine functions. We can apply these formulas to express the sum or difference of trigonometric functions sine and cosine as products and hence, simplify mathematical problems. The sum to product formulas can be derived using the product to sum formulas in trigonometry using substitutions of the variables.
In this article, we will explore the sum to product formula in detail and derive these formulas using the product to sum formulas. We will also understand the application of these formulas with the help of solved examples for a better understanding of the concept.
1.  What are Sum to Product Formulas? 
2.  Sum to Product Formulas List 
3.  Sum to Product Formulas Proof 
4.  Using Sum to Product Formula 
5.  FAQs on Sum to Product Formulas 
What are Sum to Product Formula?
The sum to product formula are used to express the sum or difference of sine function and the sum or difference of cosine function as the product of sine and cosine functions. These sum to product formulas are also known individually given by,
 Formula of sin a plus sin b, that is, sin A + sin B
 Formula of sin a minus sin b, that is, sin A  sin B
 Formula of cos a plus cos b, that is, cos A + cos B
 Formula of cos a minus cos b, that is, cos A  cos B
Please note that these formulas are not the same as the angle sum formulas in trigonometry. For example, sin A + sin B is not the same as sin (A + B). Let us now go through the formulas of the abovementioned sum to product formulas in the next section.
Sum to Product Formula List
We can prove these sum to product formulas using the product to sum formulas in trigonometry. The sum to product formula are expressed as follows:
 sin A + sin B = 2 sin [(A + B)/2] cos [(A  B)/2]
 sin A  sin B = 2 sin [(A  B)/2] cos [(A + B)/2]
 cos A  cos B = 2 sin [(A + B)/2] sin [(A  B)/2]
 cos A + cos B = 2 cos [(A + B)/2] cos [(A  B)/2]
Sum to Product Formulas Proof
Now, that we have discussed the sum to product formulas in trigonometry, let us derive these formulas using the product to sum formulas whose formulas are given by,
 sin A cos B = (1/2) [ sin (A + B) + sin (A  B) ]  (1)
 cos A sin B = (1/2) [ sin (A + B)  sin (A  B) ]  (2)
 cos A cos B = (1/2) [ cos (A + B) + cos (A  B) ]  (3)
 sin A sin B = (1/2) [ cos (A  B)  cos (A + B) ]  (4)
To derive the sum to product formulas, assume (p + q)/2 = A and (p  q)/2 = B. Now, taking the sum and difference of A and B, we have
A + B = [(p + q)/2] + [(p  q)/2]
= p/2 + q/2 + p/2  q/2
= p/2 + p/2
= p
A  B = [(p + q)/2]  [(p  q)/2]
= p/2 + q/2  p/2 + q/2
= q/2 + q/2
= q
Now, substituting the values of A, B, A + B and A  B in the formulas (1), (2), (3), and (4), we have
 sin [(p + q)/2] cos [(p  q)/2] = (1/2) [ sin p + sin q ]
2 sin [(p + q)/2] cos [(p  q)/2] = sin p + sin q  (5)  cos [(p + q)/2] sin [(p  q)/2] = (1/2) [ sin p  sin q ]
2 cos [(p + q)/2] sin [(p  q)/2] = sin p  sin q  (6)  cos [(p + q)/2] cos [(p  q)/2] = (1/2) [ cos p + cos q ]
2 cos [(p + q)/2] cos [(p  q)/2] = cos p + cos q  (7)  sin [(p + q)/2] sin [(p  q)/2] = (1/2) [ cos q  cos p ]
2 sin [(p + q)/2] sin [(p  q)/2] = cos p  cos q  (8)
Hence, we have derived the sum to product formulas which are given by the formulas (5), (6), (7) and (8).
Using Sum to Product Formula
We use the sum to product formula to simplify and solve mathematical problems in trigonometry. In this section, we will understand how to apply the sum to product formulas with the help of solving a few examples.
Example 1: Express the difference of cosines cos 4x  cos x as a product of trigonometric function using sum to product formulas.
Solution: We know that cos A  cos B = 2 sin [(A + B)/2] sin [(A  B)/2]. Substituting A = 4x and B = x into this formula, we have
cos 4x  cos x = 2 sin [(4x + x)/2] sin [(4x  x)/2]
= 2 sin (5x/2) sin (3x/2)
Answer: Hence, we can express the difference cos 4x  cos x as 2 sin (5x/2) sin (3x/2) as a product of trigonometric functions.
Example 2: Evaluate the value of sin 15° + sin 75° using the sum to product formula.
Solution: The formula required to find the value of sin 15° + sin 75° is sin A + sin B = 2 sin [(A + B)/2] cos [(A  B)/2]. Substituting A = 15° and B = 75° into the formula, we have
sin 15° + sin 75° = 2 sin [(15° + 75°)/2] cos [(15°  75°)/2]
= 2 sin(90°/2) cos (60°/2)
= 2 sin 45° cos 30°  [Because cos(x) = cos x]
= 2 × 1/√2 × √3/2  [Because cos 30° is equal to √3/2 and sin 45° is equal to 1/√2]
= √3/√2
= √(3/2)
Answer: sin 15° + sin 75 = √(3/2) using sum to product formula.
Important Notes on Sum to Product Formula
 The sum to product formulas are used to express the sum and difference of trigonometric functions sines and cosines as products of sine and cosine functions.
 We can derive the sum to product formula using the product to sum formulas in trigonometry.
 We can apply these formulas to simplify trigonometric problems.
 The sum to product formula are:
 sin A + sin B = 2 sin [(A + B)/2] cos [(A  B)/2]
 sin A  sin B = 2 sin [(A  B)/2] cos [(A + B)/2]
 cos A  cos B = 2 sin [(A + B)/2] sin [(A  B)/2]
 cos A + cos B = 2 cos [(A + B)/2] cos [(A  B)/2]
ā Related Topics:
Sum to Product Formula Examples

Example 1: Use sum to product formula to express cos 8x + cos 2x as the product.
Solution: To express cos 8x + cos 2x as the product, we will use the formula cos A + cos B = 2 cos [(A + B)/2] cos [(A  B)/2]. Substituting A = 8x and B  2x into the formula, we have
cos 8x + cos 2x = 2 cos [(8x + 2x)/2] cos [(8x  2x)/2]
= 2 cos (10x/2) cos 6x/2
= 2 cos 5x cos 3x
Answer: Hence, cos 8x + cos 2x = = 2 cos 5x cos 3x using the sum to product formula.

Example 2: Calculate the value of sin 225°  sin 135°
Solution: To find the value of sin 225°  sin 135°, we will use the sum to product formula given by, sin A  sin B = 2 sin [(A  B)/2] cos [(A + B)/2]. Substitute the values A = 225° and B = 135° into the formula, we have
sin 225°  sin 135° = 2 sin [(225°  135°)/2] cos [(225° + 135°)/2]
= 2 sin (90°/2) cos(360°/2)
= 2 sin 45° cos 180°
= 2 × 1/√2 × (1)
= √2
Answer: Hence, sin 225°  sin 135° = √2 using the sum to product formula.

Example 3: Prove that (cos 4x  cos 2x) / (sin 4x + sin 2x) =  tan x
Solution: We will use the following sum to product formulas to prove the given result:
 cos A  cos B = 2 sin [(A + B)/2] sin [(A  B)/2]
 sin A + sin B = 2 sin [(A + B)/2] cos [(A  B)/2]
Using the above formulas, we have
LHS = (cos 4x  cos 2x) / (sin 4x + sin 2x)
= 2 sin [(4x + 2x)/2] sin [(4x  2x)/2] / 2 sin [(4x + 2x)/2] cos [(4x  2x)/2]
=  [ sin (6x/2) sin (2x/2) ] / [ sin (6x/2) cos (2x/2)]
=  (sin 3x sin x) / (sin 3x cos x)
=  sin x / cos x
=  tan x
= RHS
Answer: Hence, we have proved (cos 4x  cos 2x) / (sin 4x + sin 2x) =  tan x using sum to product formulas.
FAQs on Sum to Product Formula
What are Sum to Product Formula in Trigonometry?
The sum to product formula in trigonometry are formulas that are used to express the sum and difference of sines and cosines as products of sine and cosine functions. These sum to product formula are also known individually given by,
 Formula of sin a plus sin b, that is, sin A + sin B
 Formula of sin a minus sin b, that is, sin A  sin B
 Formula of cos a plus cos b, that is, cos A + cos B
 Formula of cos a minus cos b, that is, cos A  cos B
List all Sum to Product Formula.
The sum to product formulas are expressed as follows:
 sin A + sin B = 2 sin [(A + B)/2] cos [(A  B)/2]
 sin A  sin B = 2 sin [(A  B)/2] cos [(A + B)/2]
 cos A  cos B = 2 sin [(A + B)/2] sin [(A  B)/2]
 cos A + cos B = 2 cos [(A + B)/2] cos [(A  B)/2]
How Do You Prove Sum to Product Formulas?
The sum to product formulas can be derived using the product to sum formulas in trigonometry using substitutions of the variables. We can assume (p + q)/2 = A and (p  q)/2 = B and substitute these in the product to sum formulas given by,
 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) ]
How to Use Sum to Product Formula?
The sum to product formulas are used to express the sum and difference of sine and cosine functions as products of trigonometric functions sine and cosine.
How Do You Convert Sum to Product in Trigonometry?
We can convert the sum of sine and cosine into the product of sine and cosine in trigonometry by taking appropriate assumptions of variables and substituting them into the formulas.
What is the Difference Between Product to Sum and Sum to Product Formula?
In sum to product formula, we express the sum of trigonometric functions sine and cosine as products. On the other hand, in producttosum formula, we express the product of trigonometric functions sine and cosine as the sum or difference of sine or cosine.
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