Question

Sin2x = cos2x + 2cos²x​

Answer 1

Формулы синуса и косинуса двойного угла:

$\sin2x=2\sin x\cos x$

$\cos2x=\cos^2x-\sin^2x$

Воспользуемся данными соотношениями:

$\sin2x=\cos2x+2\cos^2x$

$2\sin x\cos x=\cos^2x-\sin^2x+2\cos^2x$

$2\sin x\cos x=3\cos^2x-\sin^2x$

Перенесем все слагаемые в левую часть:

$\sin^2x+2\sin x\cos x-3\cos^2x=0$

Разделим уравнение почленно на $\cos^2x\neq 0$:

$\dfrac{\sin^2x}{\cos^2x} +\dfrac{2\sin x\cos x}{\cos^2x} -\dfrac{3\cos^2x}{\cos^2x} =0$

$\mathrm{tg}\,^2x+2\,\mathrm{tg}\,x-3=0$

Решаем квадратное уравнение относительно тангенса. Так как сумма коэффициентов уравнения равна 0, то первый корень равен 1, а второй равен свободному члену:

$\mathrm{tg}\,x_1=1\Rightarrow \boxed{x_1=\dfrac{\pi }{4} +\pi n,\ n\in\mathbb{Z}}$

$\mathrm{tg}\,x_2=-3\Rightarrow x_2=\mathrm{arctg}\,(-3) +\pi n\Rightarrow \boxed{x_2=-\mathrm{arctg}\,3 +\pi n,\ n\in\mathbb{Z}}$

Answer 2

To solve the equation sin(2x) = cos(2x) + 2cos²(x), we'll use trigonometric identities to simplify the equation. Let's start by manipulating the right-hand side of the equation.

Using the double-angle identity for cosine, cos(2x) = 1 - 2sin²(x). We can substitute this into the equation:

sin(2x) = 1 - 2sin²(x) + 2cos²(x)

Next, we'll use the Pythagorean identity, sin²(x) + cos²(x) = 1, to simplify further. Rearranging, we get sin²(x) = 1 - cos²(x). Substituting this into the equation:

sin(2x) = 1 - 2(1 - cos²(x)) + 2cos²(x)

Expanding and simplifying:

sin(2x) = 1 - 2 + 2cos²(x) + 2cos²(x) sin(2x) = -1 + 4cos²(x)

Now, the equation is simplified to sin(2x) = -1 + 4cos²(x).

To solve for x, we can use the double-angle identity for sine, which states that sin(2x) = 2sin(x)cos(x). Substituting this into the equation:

2sin(x)cos(x) = -1 + 4cos²(x)

Now, we have a quadratic equation in terms of cos(x). Rearranging and simplifying:

4cos³(x) - 2cos(x) + 1 = 0

This equation can be solved for cos(x), and once we have the value(s) of cos(x), we can find the corresponding values of x using the inverse cosine function.

Note: The solutions to this equation may involve numerical approximation or trigonometric identities, depending on the specific values of x.