Question

Сколько будет тангенс квадрат альфа + котангенс квадрат альфа, если tg альфа - ctg альфа = -4?

Answer 1

$tg \alpha - ctg\alpha = -4 \\ tg^{2} \alpha + ctg^{2} \alpha = (tg \alpha - ctg\alpha)^{2} + 2*tg \alpha*ctg\alpha = \\ = (-4)^{2} + 2*1 = 16 + 2 = 18$

Answer 2

Calculating Tangent and Cotangent Squared of Alpha

To calculate the value of tangent squared of alpha and cotangent squared of alpha when the difference between tangent and cotangent of alpha is -4, we can use the following trigonometric identities:

1. Tangent Squared Identity: \[ \tan^2(\alpha) = \sec^2(\alpha) - 1 \]

2. Cotangent Squared Identity: \[ \cot^2(\alpha) = \csc^2(\alpha) - 1 \]

Given that the difference between tangent and cotangent of alpha is -4, we can express this as: \[ \tan(\alpha) - \cot(\alpha) = -4 \]

Let's solve for tangent squared of alpha and cotangent squared of alpha using the provided information.

Solution:

We'll start by expressing the given difference in terms of sine and cosine, and then use the identities mentioned above to find the values of tangent squared and cotangent squared.

Given: \[ \tan(\alpha) - \cot(\alpha) = -4 \]

We know that: \[ \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \] \[ \cot(\alpha) = \frac{\cos(\alpha)}{\sin(\alpha)} \]

Substituting these into the given equation: \[ \frac{\sin(\alpha)}{\cos(\alpha)} - \frac{\cos(\alpha)}{\sin(\alpha)} = -4 \]

Multiplying through by \(\sin(\alpha) \cdot \cos(\alpha)\) to clear the fractions: \[ \sin^2(\alpha) - \cos^2(\alpha) = -4 \cdot \sin(\alpha) \cdot \cos(\alpha) \]

Now, we can use the trigonometric identities to express \(\sin^2(\alpha)\) and \(\cos^2(\alpha)\) in terms of \(\tan^2(\alpha)\) and \(\cot^2(\alpha)\).

Using the identity: \[ \tan^2(\alpha) + 1 = \sec^2(\alpha) \] \[ \cot^2(\alpha) + 1 = \csc^2(\alpha) \]

We can rewrite the equation in terms of \(\tan^2(\alpha)\) and \(\cot^2(\alpha)\): \[ (\tan^2(\alpha) + 1) - (\cot^2(\alpha) + 1) = -4 \cdot \sin(\alpha) \cdot \cos(\alpha) \]

Simplifying: \[ \tan^2(\alpha) - \cot^2(\alpha) = -4 \cdot \sin(\alpha) \cdot \cos(\alpha) \]

This equation gives us the relationship between the squares of the tangent and cotangent of alpha. We can use this relationship to find the values of tangent squared of alpha and cotangent squared of alpha.

Let's proceed with the calculation using the given information and the derived equation.