Sum to Product Formulae for JEE

In trigonometry, there is a particular case where certain products can be expressed as the trigonometric sum of two trigonometric expressions and vice versa. While the sum-to-product formula is derived from the multiplication and division of sin and cosine functions, the product-to-sum formula implements addition and subtraction to develop the same procedures. The chief area of Mathematics where these formulae are the most likely to be implemented is in the field of calculus, especially for integrations. In some instances, the product addition of some specific fundamental trigonometric functions can often simplify a problem further. Before we delve further into the topic, we must first understand how these formulae are derived.

Derivation of Sum to Product Formula

The sum-to-product formula allows one to represent a sum of certain trigonometric terms like sin and cosine as the products of other trigonometric functions, also termed the sum of products. It is valid to ask what the sum to the product formula is exactly. Let us take the following example to understand this concept better. 

$\sin a \cos b=\dfrac{1}{2}\{\sin (a+b)+\sin (a-b)\} \ldots 1$ 

$\cos a \sin b=\dfrac{1}{2}\{\sin (a+b)-\sin (a-b)\} \ldots 2$ 

$\cos a \cos b=\dfrac{1}{2}\{\cos (a+b)+\cos (a-b)\} \ldots 3$ 

$\sin a \sin b=\dfrac{1}{2}\{\cos (a-b)-\cos (a+b)\} \ldots 4$

If we take $\dfrac{p+q}{2}=a, \dfrac{p-q}{2}=b$

$\begin{align} &a+b=\dfrac{p+q}{2}+\dfrac{p-q}{2} \\ &a+b=p \\ &a-b=\dfrac{p+q}{2}-\dfrac{p-q}{2} \\ &a-b=q \end{align}$

Substitute the value in the above equation, and we get

$\begin{align} &\sin p+\sin q=2 \sin \left\{\left(\dfrac{p+q}{2}\right)\right\} \cos \left\{\left(\dfrac{p-q}{2}\right)\right\} \\ &\sin p-\sin q=2 \sin \left\{\left(\dfrac{p-q}{2}\right)\right\} \cos \left\{\left(\dfrac{p+q}{2}\right)\right\} \\ &\cos p+\cos q=-2 \sin \left\{\left(\dfrac{p+q}{2}\right)\right\} \sin \left\{\left(\dfrac{p-q}{2}\right)\right\} \\ &\cos p-\cos q=2 \cos \left\{\left(\dfrac{p+q}{2}\right)\right\} \cos \left\{\left(\dfrac{p-q}{2}\right)\right\} \end{align}$

As can be seen from the above example, the sum to product formula can also be drawn up for the formula of ‘sin a’ and ‘sin b’ as well as for ‘cos a’ and ‘cos b’ formulas. The most commonly used sum-to-product formulas that are extensively used in Mathematics are as follows:

$\begin{align}&\sin a+\sin b=2 \sin \left\{\left(\dfrac{a+b}{2}\right)\right\} \cos \left\{\left(\dfrac{a-b}{2}\right)\right\} \\ &\sin a-\sin b=2 \sin \left\{\left(\dfrac{a-b}{2}\right)\right\} \cos \left\{\left(\dfrac{a+b}{2}\right)\right\} \\ &\cos a+\cos b=2 \cos \left\{\left(\dfrac{a+b}{2}\right)\right\} \cos \left\{\left(\dfrac{a-b}{2}\right)\right\} \\ &\cos a-\cos b=-2 \sin \left\{\left(\dfrac{a+b}{2}\right)\right\} \sin \left\{\left(\dfrac{a-b}{2}\right)\right\} \end{align}$

Now, let us see the product to sum formulae in trigonometry. 

Derivation of the Product to Sum Formula

Before we dive deeper into the product-to-sum formulae, we will first see how each of the formulae can be derived, including the ‘cos a sin b’ formula. By using the above mentioned four equations, we can derive all the formulas related to the product of sum examples and theory that gives you a better understanding of what is the sum of product(s).

The sum and difference formulas of trigonometry are given as follows:

• sin (a + b) = sin a cos b + cos a sin b . . .(5)

• sin (a – b) = sin a cos b – cos a sin b . . .(6)

• cos (a + b) = cos a cos b – sin a sin b . . . .(7) 

• cos (a - b) =  cos a cos b + sin a sin b . . . (8)

• Adding equations (5) and (6) and then dividing by 2, we obtain 

$\sin a \cos b=\dfrac{1}{2}[\sin (a+b)+\sin (a-b)]$

• By subtracting equation (6) from (5) and then dividing by 2, we get 

$\cos a \sin b=\dfrac{1}{2}[\sin (a+b)-\sin (a-b)]$

• Adding equations (7) and (8) and dividing both sides by 2, we get $\cos a \cos b=\dfrac{1}{2}[\cos (a+b)+\cos (a-b)]$

• Subtracting equation (7) from (8) and then dividing both sides by 2, we get $\sin a \sin b=\dfrac{1}{2}[\cos (a-b)-\cos (a+b)]$

Examples Related to Product to Sum and Sum to Product Formula

Example 1: Write $\cos (3a) \cos (5a)$ as either a sum or a difference.

Solution:

As per the formula of products of sum we know that 

$\begin{align}  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\dfrac{1}{2}[\cos (2a)+\cos (8a)] \\ \end{align}$

Thus, as we can see the product of sum example is solved as follows:

$\cos (3a)\cos (5a)=\dfrac{1}{2}[\cos (2a)+\cos (8a)]$

Example 2: Express the following product of cosines as a sum:

 $2\cos \left( \dfrac{7x}{2} \right)\cos \left( \dfrac{3x}{2} \right)$.

Solution: 

Given trigonometric expression $2\cos \left( \dfrac{7x}{2} \right)\cos \left( \dfrac{3x}{2} \right)$.

According to the product to sum formula,

$\cos a\cos b=\dfrac{1}{2}[\cos (a+b)+\cos (a-b)]$.

Therefore, substituting the terms in the formula with those of the example, we get $\begin{align} & 2\cos \left( \dfrac{7x}{2} \right)\cos \left( \dfrac{3x}{2} \right) \\ & =\cos \left( \dfrac{4x}{2} \right)+\cos \left( \dfrac{10x}{2} \right) \\  & =\cos 2x+\cos 5x \\ \end{align}$

Thus, the required answer is $2 \cos \left(\dfrac{7 x}{2}\right) \cos \left(\dfrac{3 x}{2}\right)=\cos 2 x+\cos 5 x$.

Conclusion

Trigonometric functions are interrelated, and a trigonometrical product formula can be represented as a sum or difference of two other terms with sin and cosine terms. Similarly, a total or difference of two different functions can also be expressed in the form of a product. This relationship holds throughout, and if the value of one is given, the other can be figured out quickly. Specific formulas closely link the sum to product and product to sum transformations, as seen in this field of Mathematics. 

In trigonometry, the sum and difference of sines and cosines are expressed as products of sine and cosine functions using the sum to product formula. These formulas can simplify mathematical difficulties by saying the sum or difference of trigonometric functions sine and cosine as products. The product-to-sum formula in trigonometry can be used to generate the sum-to-product formula by substituting the variables.