Question

Ctg x - cos x * ctg x - (1-cos x) = 0

Answer 1

Ответ:

..............................

Answer 2

To solve the equation $ctg(x) - \cos(x) \cdot ctg(x) - (1 - \cos(x)) = 0$, let's simplify the expression:

First, distribute the $- \cos(x) \cdot ctg(x)$ term:

$ctg(x) - \cos(x) \cdot ctg(x) - 1 + \cos(x) = 0$

Next, combine like terms:

$(1 - \cos(x)) \cdot ctg(x) + \cos(x) - 1 = 0$

Now, let's solve for $ctg(x)$:

$(1 - \cos(x)) \cdot ctg(x) = 1 - \cos(x)$

Divide both sides of the equation by $(1 - \cos(x))$:

$ctg(x) = \frac{{1 - \cos(x)}}{{1 - \cos(x)}}$

Simplifying the right-hand side:

$ctg(x) = 1$

The equation simplifies to $ctg(x) = 1$. Therefore, $x$ can take values where the cotangent of $x$ is equal to 1. In other words, $x$ can be any angle whose cotangent is 1. The cotangent of an angle is equal to the reciprocal of the tangent of that angle.

Since the tangent of an angle is equal to the ratio of the sine of the angle to the cosine of the angle, we can rewrite the equation as:

$\frac{{\cos(x)}}{{\sin(x)}} = 1$

Multiplying both sides by $\sin(x)$:

$\cos(x) = \sin(x)$

Using the Pythagorean identity $\sin^2(x) + \cos^2(x) = 1$, we can rewrite $\cos(x) = \sin(x)$ as:

$\sin^2(x) + \sin^2(x) = 1$

$2\sin^2(x) = 1$

$\sin^2(x) = \frac{1}{2}$

Taking the square root of both sides:

$\sin(x) = \pm \frac{1}{\sqrt{2}}$

From the unit circle, we know that $\sin(\frac{\pi}{4}) = \frac{1}{\sqrt{2}}$ and $\sin(\frac{5\pi}{4}) = -\frac{1}{\sqrt{2}}$.

Therefore, the solutions to the equation $ctg(x) - \cos(x) \cdot ctg(x) - (1 - \cos(x)) = 0$ are:

$x = \frac{\pi}{4} + k\pi$, where $k$ is an integer.

or

$x = \frac{5\pi}{4} + k\pi$, where $k$ is an integer.