# Prove the following identity: cos x + cos 2x + cos 3x = cos 2x(1 + 2cos x)

justaguide | College Teacher | (Level 2) Distinguished Educator

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We have to prove : cos x + cos 2x + cos 3x = cos 2x(1 + 2cos x)

We know that cos A + cos B = 2 * cos (A + B)/2 * cos (A − B)/2

Now cos x + cos 2x + cos 3x

=> 2 * cos (4x / 2) * cos (2x / 2) + cos 2x

=> 2 cos 2x * cos x + cos 2x

=> cos 2x ( 1 + 2 cos x)

Therefore we prove that

cos x + cos 2x + cos 3x = cos 2x(1 + 2cos x)

giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll remove the brackets from the right side:

cos x + cos 2x + cos 3x = cos 2x + 2cos x*cos 2x

We'll eliminate the term cos 2x:

cos x + cos 3x = 2cos x*cos 2x

We'll write cos 2x = 2(cos x)^2 - 1

cos 3x = 4(cos x)^3 - 3cos x

We'll substitute them in identity and we'll get:

cos x +  4(cos x)^3 - 3cos x = 2cos x*[2(cos x)^2 - 1]

We'll combine like terms from left side and we'll remove the brackets form the right side:

4(cos x)^3 - 2cos x = 4(cos x)^3 - 2cos x q.e.d.