Question

1) Cos4α+Sin²2α 2) Sin2α+(Sinα-Cosα)²

Answer 1

Ответ:

Упростите тригонометрические выражения.

Answer 2

Let's simplify the given trigonometric expressions:

1. Cos^4(α) + Sin^2(2α):

We can use the double-angle identity for sine, which is: Sin(2α) = 2 * Sin(α) * Cos(α)

Substitute this in the expression:

Cos^4(α) + Sin^2(2α) = Cos^4(α) + (2 * Sin(α) * Cos(α))^2 = Cos^4(α) + 4 * Sin^2(α) * Cos^2(α)

Now, we can use the Pythagorean identity for sine and cosine:

Sin^2(α) + Cos^2(α) = 1

Therefore, we can express Sin^2(α) as 1 - Cos^2(α):

Cos^4(α) + 4 * (1 - Cos^2(α)) * Cos^2(α)

Now distribute the 4:

Cos^4(α) + 4 * Cos^2(α) - 4 * Cos^4(α)

Combine like terms:

-3 * Cos^4(α) + 4 * Cos^2(α)

This is the simplified expression for Cos^4(α) + Sin^2(2α).

2. Sin^2(α) + (Sin(α) - Cos(α))^2:

Expand the squared term:

(Sin(α) - Cos(α))^2 = Sin^2(α) - 2 * Sin(α) * Cos(α) + Cos^2(α)

Now, we can use the Pythagorean identity for sine and cosine:

Sin^2(α) + Cos^2(α) = 1

Therefore, we can simplify the squared term to:

(Sin(α) - Cos(α))^2 = Sin^2(α) - 2 * Sin(α) * Cos(α) + 1

Now, we can rewrite the original expression:

Sin^2(α) + (Sin(α) - Cos(α))^2 = Sin^2(α) + Sin^2(α) - 2 * Sin(α) * Cos(α) + 1

Combine like terms:

2 * Sin^2(α) - 2 * Sin(α) * Cos(α) + 1

This is the simplified expression for Sin^2(α) + (Sin(α) - Cos(α))^2.