Question

Sin^8(13π/12)-cos^8(11π/12)

Answer 1

Ответ:  во вложении Объяснение:

Answer 2

Ответ:приложено

Объяснение:

Answer 3

To evaluate the expression sin^8(13π/12) - cos^8(11π/12), we can use trigonometric identities to simplify the expression.

Let's start by expressing the angles in terms of common angles:

13π/12 = π/12 + π 11π/12 = π/12 + π/2

Using the trigonometric identities:

sin(x + π) = -sin(x) cos(x + π) = -cos(x) sin^2(x) + cos^2(x) = 1

We can rewrite the expression as:

sin^8(13π/12) - cos^8(11π/12) = (-sin(π/12))^8 - (-cos(π/12 + π/2))^8 = sin^8(π/12) - cos^8(π/12) = (sin^2(π/12))^4 - (cos^2(π/12))^4 = [(1 - cos^2(π/12))^2]^4 - (cos^2(π/12))^4 = [1 - 2cos^2(π/12) + (cos^2(π/12))^2]^4 - (cos^2(π/12))^4

Now, let's substitute cos^2(π/12) with a variable for simplicity. Let's say u = cos^2(π/12).

= (1 - 2u + u^2)^4 - u^4 = u^8 - 8u^7 + 28u^6 - 56u^5 + 70u^4 - 56u^3 + 28u^2 - 8u + 1 - u^4 = u^8 - 8u^7 + 28u^6 - 55u^5 + 69u^4 - 56u^3 + 28u^2 - 8u + 1

Now, substitute back u with cos^2(π/12):

= cos^16(π/12) - 8cos^14(π/12) + 28cos^12(π/12) - 55cos^10(π/12) + 69cos^8(π/12) - 56cos^6(π/12) + 28cos^4(π/12) - 8cos^2(π/12) + 1

This is the simplified expression for sin^8(13π/12) - cos^8(11π/12).