Question

1+tgx=2tg^2 выручайте мужики ​

Answer 1

Ответ:

$x_{1} = - arctg \frac{1}{2} + \pi \: n \\ x_{2} = \frac{\pi}{4} + \pi \: n$

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Пошаговое объяснение:

$1 + tgx = 2 {tg}^{2} x \\ 2 {tg}^{2} x - tgx - 1 = 0$

тригонометрическое квадратное уравнение, замена переменной:

tgx=y

$2 {y}^{2} - y - 1 = 0 \\ y_{1} = - \frac{1}{2} \\ y_{2} = 1$

обратная замена:

$y_{1} = - \frac{1}{2} \\ tgx = - \frac{1}{2} \\ x = arctg( - \frac{1}{2} ) + \pi \: n$

n €Z, знак € читать принадлежит

$x = - arctg \frac{1}{2} + \pi \: n$

$y_{2} = 1 \\ tgx = 1 \\ x = arctg1 + \pi \: n \\ x = \frac{\pi}{4} + \pi \: n$

Answer 2

Solving the Equation 1tgx = 2tg^2

To solve the equation 1tgx = 2tg^2, we can use trigonometric identities and properties to simplify and find the value of x.

First, let's rewrite the equation using the tangent function:

1tgx = 2tg^2

The tangent function can be expressed as the ratio of the sine and cosine functions:

tan(x) = sin(x) / cos(x)

Using this identity, we can rewrite the equation in terms of sine and cosine:

sin(x) / cos(x) = 2 * (sin(x) / cos(x))^2

Now, we can solve for x by manipulating the equation using trigonometric identities and properties.

Let's proceed with the solution.