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

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Answer 1 · 2018-04-17T07:50:13

See explanation

We want to verify the identity

$cos (θ) + cos (2 θ) + cos (3 θ) = (2 cos (θ) + 1) cos (2 θ)$ $cos(theta)+cos(2theta)+cos(3theta)=(2cos(theta)+1)cos(2theta)$

We will use the identity

$cos (a) cos (b) = \frac{1}{2} (cos (a - b) + cos (a + b))$ $cos(a)cos(b)=1/2(cos(a-b)+cos(a+b))$

Thus

$R H S = (2 cos (θ) + 1) cos (2 θ)$ $RHS=(2cos(theta)+1)cos(2theta)$

$L H S = 2 cos (2 θ) cos (θ) + cos (2 θ)$ $color(white)(LHS)=2cos(2theta)cos(theta)+cos(2theta)$

$L H S = 2 (\frac{1}{2} ((cos (2 θ - θ) + cos (2 θ + θ)))) + cos (2 θ)$ $color(white)(LHS)=2(1/2((cos(2theta-theta)+cos(2theta+theta))))+cos(2theta)$

$L H S = cos (2 θ - θ) + cos (2 θ + θ) + cos (2 θ)$ $color(white)(LHS)=cos(2theta-theta)+cos(2theta+theta)+cos(2theta)$

$L H S = cos (θ) + cos (3 θ) + cos (2 θ) = L H S$ $color(white)(LHS)=cos(theta)+cos(3theta)+cos(2theta)=LHS$