Question

Cos2x*sinx+sin2x*cosx

Answer 1

Полное решение задания в файле.

Answer 2

To simplify the expression $\cos(2x) \sin(x) + \sin(2x) \cos(x)$, you can use trigonometric identities. One way to simplify it is by using the product-to-sum identities, which are:

1. $\sin(A) \cos(B) = \frac{1}{2} \left[ \sin(A + B) + \sin(A - B) \right]$
2. $\cos(A) \sin(B) = \frac{1}{2} \left[ \sin(A + B) - \sin(A - B) \right]$

Let's apply these identities to your expression:

$$\begin{align*} \cos(2x) \sin(x) + \sin(2x) \cos(x) &= \frac{1}{2} \left[ \sin(2x + x) + \sin(2x - x) \right] + \frac{1}{2} \left[ \sin(2x + x) - \sin(2x - x) \right] \\ &= \frac{1}{2} \left[ \sin(3x) + \sin(x) \right] + \frac{1}{2} \left[ \sin(3x) - \sin(x) \right] \\ &= \frac{1}{2} \sin(3x) + \frac{1}{2} \sin(x) + \frac{1}{2} \sin(3x) - \frac{1}{2} \sin(x) \\ &= \frac{1}{2} \sin(3x) + \frac{1}{2} \sin(3x) \\ &= \sin(3x). \end{align*}$$

So, $\cos(2x) \sin(x) + \sin(2x) \cos(x) = \sin(3x)$.