Question

Sin 4 альфа/ 2 cos 2 альфа

Answer 1

Ответ:

$\frac{ \sin( 4\alpha ) }{ 2\cos(2 \alpha ) } = \frac{2 \sin( 2\alpha ) \cos( 2\alpha ) }{ 2\cos( 2\alpha ) } = \sin(2 \alpha )$

Answer 2

To simplify the expression $\frac{\sin(4\alpha)}{2\cos(2\alpha)}$, we can use trigonometric identities.

First, let's start with the double-angle identity for sine:

$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$

We can rewrite $\sin(4\alpha)$ as:

$\sin(4\alpha) = 2\sin(2\alpha)\cos(2\alpha)$

Now, we can substitute this into the original expression:

$\frac{\sin(4\alpha)}{2\cos(2\alpha)} = \frac{2\sin(2\alpha)\cos(2\alpha)}{2\cos(2\alpha)}$

Now, the $\cos(2\alpha)$ terms cancel out:

$\frac{2\sin(2\alpha)\cos(2\alpha)}{2\cos(2\alpha)} = \frac{2\sin(2\alpha)\cancel{\cos(2\alpha)}}{2\cancel{\cos(2\alpha)}}$

This simplifies to:

$\sin(2\alpha)$

So, $\frac{\sin(4\alpha)}{2\cos(2\alpha)} = \sin(2\alpha)$.