Prove the following identity: cos x + cos 2x + cos 3x = cos 2x(1 + 2cos x)
| Certified Educator
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)
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.
Access hundreds of thousands of answers with a free trial.
- Prove the following identity: (1+ sin x + cos x) / (1+ sin x - cos x) = cot x/2
- 1 educator answer
- Prove the following identity: cos4x - sin4xcot2x = -1
- 1 educator answer
- What is x if cos^2(x/2)-2cos^2x=(3/2)*square root2(1+cosx)+2sin^2x?
- 1 educator answer
- Prove the identity cotx*sinx=cosx/(cos^2x+sin^2x)
- 1 educator answer
- Prove the identity (3secx+3cosx)(3secx-3cosx)=9(tan^2x+sin^2x).
- 1 educator answer
- How do we derive the expansion for sin 3x?
- 1 educator answer
- What is the angle x if the identity sin2x-(cos 2x)/2 -1/2=cosx-2cosx*tanx?
- 1 educator answer
- how to evaluate cos13pi/6 and sin37pi/4?
- 1 educator answer
- finding an anglewhat is the x angle if 2sin2x+square root 2=0?
- 1 educator answer
- What is x if (1+cos2x)/2=sin(-2x)?
- 1 educator answer
