Question

1) (1+ctg^2альфа)•sin^2альфа-cos^2 альфа 2) (1+tg^2alfa)•cos^2alfa-sin^2alfa

Answer 1

Вот ответ на твой вопрос.

Answer 2

To simplify the given expressions, we'll use trigonometric identities to rewrite them in a more manageable form.

1. Simplifying (1+ctg^2α)•sin^2α - cos^2α:

First, we'll express cotangent (ctg) in terms of sine and cosine:

cot(α) = 1/tan(α) = cos(α)/sin(α)

Now, let's rewrite the expression:

(1 + cot^2(α))•sin^2(α) - cos^2(α)

Substitute cot(α):

(1 + (cos(α)/sin(α))^2)•sin^2(α) - cos^2(α)

Simplify the cotangent expression:

(1 + cos^2(α)/sin^2(α))•sin^2(α) - cos^2(α)

Now, we can combine the terms:

(sin^2(α) + cos^2(α))•sin^2(α) - cos^2(α)

Using the Pythagorean identity sin^2(α) + cos^2(α) = 1:

(1)•sin^2(α) - cos^2(α)

Finally, the simplified expression is:

sin^2(α) - cos^2(α)

2. Simplifying (1+tg^2α)•cos^2α - sin^2α:

First, we'll express tangent (tg) in terms of sine and cosine:

tan(α) = sin(α)/cos(α)

Now, let's rewrite the expression:

(1 + tan^2(α))•cos^2(α) - sin^2(α)

Substitute tan(α):

(1 + (sin(α)/cos(α))^2)•cos^2(α) - sin^2(α)

Simplify the tangent expression:

(1 + sin^2(α)/cos^2(α))•cos^2(α) - sin^2(α)

Now, we can combine the terms:

(cos^2(α) + sin^2(α))•cos^2(α) - sin^2(α)

Using the Pythagorean identity sin^2(α) + cos^2(α) = 1:

(1)•cos^2(α) - sin^2(α)

Finally, the simplified expression is:

cos^2(α) - sin^2(α)

In both cases, we end up with expressions that involve trigonometric identities and are not further reducible without knowing the specific value of α.