Question

Sin2xsinx-cos2xcosx1/2

Answer 1

Ответ:

$\sin(2x) \sin(x) - \cos(2x) \cos(x) \leqslant \frac{1}{2} \\ - ( \cos(2x) \cos(x) - \sin(2x) \sin(x) ) \leqslant \frac{1}{2} \\ \cos(2x + x) \geqslant - \frac{1}{2} \\ \cos(3x) \geqslant - \frac{1}{2}$

рисунок

$3x\in[ - \frac{2\pi}{3} + 2\pi \: n; \frac{2\pi}{3} + 2\pi \: n] \\ x\in[- \frac{2\pi}{9} + \frac{2\pi \: n}{3} ; \frac{2\pi}{9} + \frac{2\pi \: n}{3} ] \\ n\in\:Z$

Answer 2

To simplify the expression sin(2x)sin(x) - cos(2x)cos(x), you can use trigonometric identities.

First, you can use the product-to-sum identities:

1. sin(2x)sin(x) = 1/2 * [cos(2x - x) - cos(2x + x)] = 1/2 * [cos(x) - cos(3x)]
2. cos(2x)cos(x) = 1/2 * [cos(2x - x) + cos(2x + x)] = 1/2 * [cos(x) + cos(3x)]

Now, the expression becomes:

(1/2 * [cos(x) - cos(3x)]) - (1/2 * [cos(x) + cos(3x)])

Now, you can simplify further:

1/2 * [cos(x) - cos(3x)] - 1/2 * [cos(x) + cos(3x)]

= 1/2 * cos(x) - 1/2 * cos(3x) - 1/2 * cos(x) - 1/2 * cos(3x)

= (1/2 * cos(x) - 1/2 * cos(x)) - (1/2 * cos(3x) + 1/2 * cos(3x))

= 0 - 1 * cos(3x)

= -cos(3x)

So, sin(2x)sin(x) - cos(2x)cos(x) simplifies to -cos(3x).