Question

1-sin^2 альфа +tg^2 альфа *cos^2 альфа

Answer 1

$1-\sin^2\alpha+{\rm tg}^2\alpha\cdot \cos^2\alpha=\cos^2\alpha+\dfrac{\sin^2\alpha}{\cos^2\alpha}\cdot \cos^2\alpha=\cos^2\alpha+\sin^2\alpha=1$

Answer 2

To simplify the expression, let's use trigonometric identities:

1. Start with the expression: 1 - sin^2(α) + tan^2(α) * cos^2(α)

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

3. Substitute sin^2(α) in the expression: 1 - sin^2(α) + tan^2(α) * cos^2(α) = 1 - (1 - cos^2(α)) + tan^2(α) * cos^2(α)

4. Simplify: 1 - 1 + cos^2(α) + tan^2(α) * cos^2(α) = cos^2(α) + tan^2(α) * cos^2(α)

5. Factor out cos^2(α) from the expression: cos^2(α) * (1 + tan^2(α))

Therefore, the simplified expression is cos^2(α) * (1 + tan^2(α)).