Product To Sum Formulas

The Product to sum formulas in trigonometry are formulas that are used to express the product of sine and cosine functions into the sum and difference of sine and cosine functions. We can apply these formulas to express the product of trigonometric functions into sum and the difference of sine and cosine functions. In real life, we use Product to sum formula to simplify the expression in trignometric functions.

In this article, we will cover the concept of Product into Sum/Difference. This category falls under the broader category of Trigonometry, which is a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination(JEE Main) and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more.

What are Product-to-sum/difference formulae?

The Product formula is used to express the Product of sine and cosine functions into the sum or difference of sine and cosine functions. The sum and difference formulas of sine and cosine functions are added or subtracted to derive these identities. The product-to-sum identities can be used to simplify the trigonometric expression.

Product into Sum/Difference Formulas

Product-to-sum formulas provide a powerful tool for simplifying trigonometric expressions involving products of sines and cosines, and the product to sum formulas are:

1. $2 \cos \alpha \cos \beta=[\cos (\alpha-\beta)+\cos (\alpha+\beta)]$
2. $2 \sin \alpha \sin \beta=[\cos (\alpha-\beta)-\cos (\alpha+\beta)]$
3. $2 \sin \alpha \cos \beta=[\sin (\alpha+\beta)+\sin (\alpha-\beta)]$
4. $2 \cos \alpha \sin \beta=[\sin (\alpha+\beta)-\sin (\alpha-\beta)]$

where, $\alpha$ and $\beta$ are two angles of a triangle
1) $2 \cos \alpha \cos \beta=\cos (\alpha-\beta)+\cos (\alpha+\beta)$

This formula involves the conversion of the product of cosine functions of two different angles into a sum of the cosine angle.
2) $2 \sin \alpha \sin \beta=\cos \left(\alpha-\beta\right)-\cos \left(\alpha+\beta\right)$

This formula involves the conversion of the product of sine functions of two different angles into a difference in the cosine angle.
3) $2 \sin \alpha \cos \beta=\sin (\alpha+\beta)+\sin (\alpha-\beta)$

This formula involves the conversion of the product of sine and cosine functions of two different angles into a sum of the sine angle.
4) $2 \cos \alpha \sin \beta=\sin \left(\alpha{+} \beta\right)-\sin (\alpha-\beta)$

This formula involves the conversion of the product of sine and cosine functions of two different angles into a different of sine angle.

Proof of Product-to-sum formulae:

We can derive the product-to-sum formula from the sum and difference identities

Product of cosines

$
\begin{array}{r}
\cos \alpha \cos \beta+\sin \alpha \sin \beta=\cos (\alpha-\beta) \\
+\cos \alpha \cos \beta-\sin \alpha \sin \beta=\cos (\alpha+\beta) \\
\hline 2 \cos \alpha \cos \beta=\cos (\alpha-\beta)+\cos (\alpha+\beta)
\end{array}
$

Product of sine and cosine

$
\begin{gathered}
\sin (\alpha+\beta)=\sin (\alpha) \cos (\beta)+\cos (\alpha) \sin (\beta) \\
+\sin (\alpha-\beta)=\sin (\alpha) \cos (\beta)-\cos (\alpha) \sin (\beta) \\
\hline \sin (\alpha+\beta)+\sin (\alpha-\beta)=2 \sin (\alpha) \cos (\beta)
\end{gathered}
$

Product of cosine

$
\begin{gathered}
\cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta \\
-\quad \cos (\alpha+\beta)=-(\cos \alpha \cos \beta-\sin \alpha \sin \beta) \\
\hline \cos (\alpha-\beta)-\cos (\alpha+\beta)=2 \sin \alpha \sin \beta
\end{gathered}
$

Solved Example Based on Product into Sum/Difference

Example 1: The value of $
\cos \left(\frac{2 \pi}{7}\right)+\cos \left(\frac{4 \pi}{7}\right)+\cos \left(\frac{6 \pi}{7}\right)
$ is equal to? [JEE MAINS 2022]
Solution:

Using Summation of cosine series

$
\begin{aligned}
\cos \frac{2 \pi}{7}+\cos \frac{4 \pi}{7}+\cos \frac{6 \pi}{7} & =\frac{\sin 3 \times \frac{2 \pi}{2 \times 7}}{\sin \frac{2 \pi}{2 \times 7}} \times \cos \left(\frac{\frac{2 \pi}{7}+\frac{6 \pi}{7}}{2}\right)
\end{aligned} $

$\begin{aligned} & =\frac{\sin \frac{5 \pi}{7}}{\sin \frac{\pi}{7}} \times \cos \frac{4 \pi}{7} \\ & =\frac{\sin \left(\pi-\frac{4 \pi}{7}\right) \cos \frac{4 \pi}{7}}{\sin \frac{\pi}{7}}\end{aligned}$
$
\begin{aligned}
& =\frac{2 \sin \frac{4 \pi}{7} \cos \frac{4 \pi}{7}}{2 \sin \frac{\pi}{7}} \\
& =\frac{\sin \frac{8 \pi}{7}}{2 \sin \frac{\pi}{7}}=-\frac{1}{2}
\end{aligned}
$
Hence the answer is $1 / 2$

Example 2: The value of $\cos ^2 10^{\circ}-\cos 10^{\circ} \cos 50^{\circ}+\cos ^2 50^{\circ}$ is [JEE MAINS 2019]
Solution: $\cos ^2 10^{\circ}-\cos 10^{\circ} \cos 50^{\circ}+\cos ^2 50^{\circ}$

$
\begin{aligned}
& \Rightarrow \frac{1+\cos 20^{\circ}}{2}+\frac{1+\cos 100^{\circ}}{2}-\cos 10^{\circ} \cos 50^{\circ} \\
& \Rightarrow \frac{1}{2}\left[2+\cos 20^{\circ}+\cos 100^{\circ}-2 \cos 10^{\circ} \cos 50^{\circ}\right] \\
& \Rightarrow \frac{1}{2}\left[2+\cos 100^{\circ}+\cos 20^{\circ}-\cos 60^{\circ}-\cos 40^{\circ}\right] \\
& \Rightarrow \frac{1}{2}\left[\frac{3}{2}+2 \cos 60^{\circ} \cdot \cos 40^{\circ}-\cos 40^{\circ}\right] \\
& \Rightarrow \frac{1}{2}\left[\frac{3}{2}+2 \times \frac{1}{2} \cos 40^{\circ}-\cos 40^{\circ}\right]=\frac{3}{4}
\end{aligned}
$
Hence, the answer is $3 / 4$

Example 3: If $x+\frac{1}{x}=2 \cos \theta$, then $x^3+\frac{1}{x^3}$ ?
Solution

$
\begin{aligned}
&\left(x+\frac{1}{x}\right)^3=x^3+\frac{1}{x^3}+3 x \cdot \frac{1}{x}\left(x+\frac{1}{x}\right) \quad \because(x+y)^3=x^3+y^3+3 x y(x+y) \\
& x^3+\frac{1}{x^3}=\left(x+\frac{1}{x}\right)^3-3 x \cdot \frac{1}{x}\left(x+\frac{1}{x}\right) \quad \text { Given }\left(x+\frac{1}{x}\right)=2 \cos \theta \\
& x^3+\frac{1}{x^3}=(2 \cos \theta)^3-3 \cdot 2 \cos \theta \\
&=2\left[4 \cos ^3 \theta-3 \cos \theta\right] \\
&=2 \cos 3 \theta
\end{aligned}
$
Hence, the answer is $2 \cos 3 \theta$

Example 4: The value of $\sin \left(18^{\circ}\right)+\sin \left(72^{\circ}\right)-\sqrt{2} \cos \left(27^{\circ}\right)$ is
Solution

$
\begin{aligned}
& \sin \left(18^{\circ}\right)+\sin \left(72^{\circ}\right)-\sqrt{2} \cos \left(27^{\circ}\right) \\
& =2 \sin \left(45^{\circ}\right) \cos 27^{\circ}-\sqrt{2} \cos \left(27^{\circ}\right) \\
& =0
\end{aligned}
$
Hence, the answer is 0.

Example 5: if $\sin (3 x)+\sin (2 x)-\sin (x)=0$, then find the number of solutions in $[0, \pi]$ ?
Solution

$
\Rightarrow \sin (3 x)+\sin (2 x)-\sin (x)=0
$

$
\begin{aligned}
& \Rightarrow \sin (3 x)-\sin (x)+\sin (2 x)=0 \\
& \Rightarrow 2 \sin (x) \cos (2 x)+\sin (2 x)=0 \\
& \Rightarrow 2 \sin (x) \cos (2 x)+2 \sin (x) \cos (x)=0 \\
& \Rightarrow 2 \sin (x)\{\cos (2 x)+\cos (x)=0 \\
& \Rightarrow 2 \sin (x)\left\{2 \cos \left(\frac{3 x}{2}\right) \cos \left(\frac{x}{2}\right)\right\}=0 \\
& \Rightarrow x=0, \frac{\pi}{3}, \pi
\end{aligned}
$

Hence, the answer is the 3.

Summary

The product-to-sum formulas in trigonometry are used for simplifying and transforming products of trigonometric functions into sums. These formulas are essential in various applications, including simplifying complex trigonometric expressions, solving equations, and deriving identities. Understanding and applying these formulas enhances problem-solving skills in trigonometry.