How do you prove cosθ + cos2θ + cos3θ = (2cosθ + 1) cos2θ ?

Apr 17, 2018

See explanation

Explanation:

We want to verify the identity

$\cos \left(\theta\right) + \cos \left(2 \theta\right) + \cos \left(3 \theta\right) = \left(2 \cos \left(\theta\right) + 1\right) \cos \left(2 \theta\right)$

We will use the identity

• $\cos \left(a\right) \cos \left(b\right) = \frac{1}{2} \left(\cos \left(a - b\right) + \cos \left(a + b\right)\right)$

Thus

$R H S = \left(2 \cos \left(\theta\right) + 1\right) \cos \left(2 \theta\right)$

$\textcolor{w h i t e}{L H S} = 2 \cos \left(2 \theta\right) \cos \left(\theta\right) + \cos \left(2 \theta\right)$

$\textcolor{w h i t e}{L H S} = 2 \left(\frac{1}{2} \left(\left(\cos \left(2 \theta - \theta\right) + \cos \left(2 \theta + \theta\right)\right)\right)\right) + \cos \left(2 \theta\right)$

$\textcolor{w h i t e}{L H S} = \cos \left(2 \theta - \theta\right) + \cos \left(2 \theta + \theta\right) + \cos \left(2 \theta\right)$

$\textcolor{w h i t e}{L H S} = \cos \left(\theta\right) + \cos \left(3 \theta\right) + \cos \left(2 \theta\right) = L H S$