Question

Ctg²(x-π/2)-ctg(x-3π/2)-2=0

Answer 1

${ctg}^{2} (x - \binom{\pi}{2} ) - ctg(x - \frac{3\pi}{2} ) - 2 = 0 \\ {tg}^{2} x - tg \: x - 2 = 0 \\ tg \: x = y$
${y}^{2} - y - 2 = 0 \\ y = - 1 \\ y = 2$
$tg \: x = - 1 \\ x = \pi \times n - \frac{\pi}{4} \\ tg \: x = 2\\ x = \pi \times n + ctg \: 2$

Answer 2

To solve the equation $\text{ctg}^2(x-\frac{\pi}{2}) - \text{ctg}(x-\frac{3\pi}{2}) - 2 = 0$, we can use some trigonometric identities and algebraic manipulations. First, let's rewrite the equation in terms of basic trigonometric functions:

Since $\text{ctg}(x) = \frac{1}{\text{tan}(x)}$, we have: $\frac{1}{\text{tan}^2(x-\frac{\pi}{2})} - \frac{1}{\text{tan}(x-\frac{3\pi}{2})} - 2 = 0$

Next, we can use the following trigonometric identities:

1. $\text{tan}(x - \frac{\pi}{2}) = -\text{cot}(x)$
2. $\text{tan}(x - \frac{3\pi}{2}) = \text{cot}(x)$

Substituting these identities into the equation, we get: $\frac{1}{(-\text{cot}(x))^2} - \frac{1}{\text{cot}(x)} - 2 = 0$

Now, we can simplify further: $\frac{1}{\text{cot}^2(x)} - \frac{1}{\text{cot}(x)} - 2 = 0$

To make it easier to work with, let's substitute $\text{cot}(x)$ with $t$: $\frac{1}{t^2} - \frac{1}{t} - 2 = 0$

Now, let's find a common denominator and combine the fractions: $\frac{1 - t}{t^2} - 2 = 0$

Multiply both sides by $t^2$ to eliminate the fraction: $1 - t - 2t^2 = 0$

Rearrange the equation in standard quadratic form: $2t^2 + t - 1 = 0$

Now, we can solve this quadratic equation for $t$. You can use the quadratic formula: $t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For our equation, $a = 2, b = 1,$ and $c = -1$. Substituting these values into the formula, we get: $t = \frac{-(1) \pm \sqrt{(1)^2 - 4(2)(-1)}}{2(2)}$

Now, calculate the discriminant: $\text{Discriminant} = (1)^2 - 4(2)(-1) = 9$

Since the discriminant is positive, we have two distinct real solutions: $t = \frac{-1 + \sqrt{9}}{4} = \frac{-1 + 3}{4} = \frac{1}{2}$ $t = \frac{-1 - \sqrt{9}}{4} = \frac{-1 - 3}{4} = -1$

Now, recall that $t = \text{cot}(x)$:

1. $\text{cot}(x) = \frac{1}{2}$