Question

Prove the trigonometric formula: \[2\cos (a + b)\cos (a - b) = \cos 2a + \cos 2b\].

Answer 1

Hint: There are many trigonometric identities to do this question. but the easiest way to answer this question is by using the identity \[2\cos x \cos y = \cos (x + y) + \cos (x - y)\] , in which we can assume $x = a + b$ and $y = a - b$ .

Complete step-by-step solution:
There are multiple ways to do this question but we will do this by the easiest method.
In the question, we have to prove that \[2\cos (a + b)\cos (a - b) = \cos 2a + \cos 2b\]
We will use the formula \[2\cos x \cos y = \cos (x + y) + \cos (x - y)\] to do this question.
Now, let’s assume $x = a + b$ and $y = a - b$.
Put these values in above formula
\[ \Rightarrow 2\cos\left( {a + b} \right)\cos\left( {a - b} \right) = \cos(a + b + \left( {a - b} \right)) + \cos(a + b - \left( {a - b} \right))\]
\[ \Rightarrow 2\cos\left( {a + b} \right)\cos\left( {a - b} \right) = \cos(a + b + a - b) + \cos(a + b - a + b)\]
on simplifying the equation in RHS, we get
\[ \Rightarrow 2\cos\left( {a + b} \right)\cos\left( {a - b} \right) = \cos(2a) + \cos(2b)\]
Hence proved.
Additional information: Trigonometric identities are equations that relate different trigonometric functions and are true for any value of the variable that is there in the domain. Basically, an identity is an equation that holds true for all the values of the variable(s) present in it.

Note: There is one method through which we can do this question. We can use the formula \[\cos2\theta = 2co{s^2}\theta - 1.\] To solve this question. First, we will expand the part written in LHS using the identity $\cos \left( {a + b} \right) = (\cos a \cos b - \sin a \sin b)$ and $\cos \left( {a - b} \right) = (\cos a \cos b + \sin a \sin b)$. Then we will use the identity $\left( {a - b} \right)\left( {a + b} \right) = \left( {{a^2} - {b^2}} \right)$ to solve this question further. Then we will use the identity ${\sin ^2}\theta + {\cos ^2}\theta = 1$ to convert all the sin function into cos function and then we will do further simplification to get into the position so that we can use the formula \[\cos2\theta = 2co{s^2}\theta - 1.\] to get to the end result.