As we know that there are six Trigonometric functions of angles and their names are:

These functions are in relation to the right triangle in the following way:

In any right triangle ABC,

Sin A = Perpendicular/ Hypotenuse

Cos A = Base/ Hypotenuse

Tan A = Perpendicular/ Base

Cot A = Base/ Perpendicular

Cosec A = Hypotenuse/Perpendicular

Sec A = Hypotenuse/ Base

Formula of 2cosacosb

We know that,

cos (A + B) = cos A cos B – sin A sin B ….. (1)

cos (A – B) = cos A cos B + sin A sin B ….. (2)

Adding (1) and (2), we get

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

Solved Examples

Example 1: Prove that

$\begin{array}{l} c o s 2 x c o s \frac{x}{2} - c o s 3 x c o s \frac{9 x}{2} = s i n 5 x s i n \frac{5 x}{2} \end{array}$

Solution:

\begin{array}{l} L H S = c o s 2 x c o s \frac{x}{2} - c o s 3 x c o s \frac{9 x}{2} \\ = \frac{1}{2} [2 c o s 2 x c o s \frac{x}{2} - 2 c o s 3 x c o s \frac{9 x}{2}] \end{array}

Using the formula 2 cos A cos B = cos (A + B) + cos (A – B),

\begin{array}{l} = \frac{1}{2} [c o s (2 x + \frac{x}{2}) + c o s (2 x - \frac{x}{2}) - c o s (\frac{9 x}{2} + 3 x) - c o s (\frac{9 x}{2} - 3 x)] \end{array}

\begin{array}{l} = \frac{1}{2} [c o s \frac{5 x}{2} + c o s \frac{3 x}{2} - c o s \frac{15 x}{2} - c o s \frac{3 x}{2}] \\ = \frac{1}{2} [c o s \frac{5 x}{2} - c o s \frac{15 x}{2}] \end{array}

Using the formula of cos x – cos y,

\begin{array}{l} = \frac{1}{2} [- 2 s i n \frac{\frac{5 x}{2} + \frac{15 x}{2}}{2} s i n \frac{\frac{5 x}{2} - \frac{15 x}{2}}{2}] \\ = - s i n 5 x s i n (\frac{- 5 x}{2}) \\ = s i n 5 x s i n \frac{5 x}{2} \\ = R H S \end{array}

Hence, proved that

\begin{array}{l} c o s 2 x c o s \frac{x}{2} - c o s 3 x c o s \frac{9 x}{2} = s i n 5 x s i n \frac{5 x}{2} \end{array}

Example 2: Express 6 cos x cos 2x in terms of sum function.
Solution:
Consider,
6 cos x cos 2x
= 3 [2 cos x cos 2x]

Using the formula 2 cos A cos B = cos (A + B) + cos (A – B),
= 3[cos (x + 2x) + cos (x – 2x)] = 3[cos 3x + cos (-x)] = 3 [cos 3x + cos x]

To learn other trigonometric formulas Register yourself at BYJU’S.