# How do you prove that? Sin4x=4 cos2x sinx cosx

Jan 7, 2017

Use algebra and trigonometric identities to change only one side of the equation, until it looks the same as the other side.

#### Explanation:

Prove: $\sin \left(4 x\right) = 4 \cos \left(2 x\right) \sin \left(x\right) \cos \left(x\right)$

I will only change the left side:

Substitute $\sin \left(2 x + 2 x\right) \text{ for } \sin \left(4 x\right)$:

$\sin \left(2 x + 2 x\right) = 4 \cos \left(2 x\right) \sin \left(x\right) \cos \left(x\right)$

Use the identity $\sin \left(A + B\right) = \sin \left(A\right) \cos \left(B\right) + \cos \left(A\right) \sin \left(B\right)$ where $A = B = 2 x$:

$\sin \left(2 x\right) \cos \left(2 x\right) + \cos \left(2 x\right) \sin \left(2 x\right) = 4 \cos \left(2 x\right) \sin \left(x\right) \cos \left(x\right)$

Combine the two terms on the left:

$2 \cos \left(2 x\right) \sin \left(2 x\right) = 4 \cos \left(2 x\right) \sin \left(x\right) \cos \left(x\right)$

Use the identity sin(2x) = 2sin(x)cos(x) on the left:

$2 \cos \left(2 x\right) 2 \sin \left(x\right) \cos \left(x\right) = 4 \cos \left(2 x\right) \sin \left(x\right) \cos \left(x\right)$

Multiply the 2s on the left:

$4 \cos \left(2 x\right) \sin \left(x\right) \cos \left(x\right) = 4 \cos \left(2 x\right) \sin \left(x\right) \cos \left(x\right)$

The left side looks the same as the right side. Q.E.D.

Jan 7, 2017

See the Proof in Explanation.

#### Explanation:

We have, $\sin 2 \theta = 2 \sin \theta \cos \theta$

Letting $\theta = 2 x ,$ we get, $\sin \left(2 \left(2 x\right)\right) = 2 \sin 2 x \cos 2 x ,$ i.e.,

$\sin 4 x = 2 \cos 2 x \sin 2 x$, and, using the same identity for $\sin 2 x$,

$\sin 4 x = \left(2 \cos 2 x\right) \left(2 \sin x \cos x\right) = 4 \cos 2 x \sin x \cos x$.