Question

2sin pi/12(cos^4 pi/24-sin^4 pi/24)

Answer 1

2 sin pi/12 ( cos^2 pi/24+ sin^2 pi/24)(cos^2 pi/24 - sin^2 pi/24) =
=2 sin pi/12 * 1 * cos (2 *pi/24) = 2 sin pi/12 * cos pi/12 = sin (2* pi/12) = sin pi/6 = 1/2 = 0,5

Answer 2

To simplify the expression 2sin(pi/12)(cos^4(pi/24) - sin^4(pi/24)), we can use trigonometric identities and simplify each term step by step.

1. Start by simplifying the expression inside the parentheses: cos^4(pi/24) - sin^4(pi/24)

   We can use the identity: cos^2(x) - sin^2(x) = cos(2x). Therefore, cos^4(x) - sin^4(x) = (cos^2(x) - sin^2(x))(cos^2(x) + sin^2(x)) = cos(2x)(1) = cos(2x)

   Applying this identity, we have: cos^4(pi/24) - sin^4(pi/24) = cos(2 * pi/24) = cos(pi/12)

2. Now substitute this simplified expression back into the original expression: 2sin(pi/12)(cos(pi/12))

3. We can use the identity: sin(2x) = 2sin(x)cos(x). Therefore, sin(pi/12) = (1/2)sin(2 * pi/12) = (1/2)sin(pi/6) = (1/2)(1/2) = 1/4.

4. Substitute the values back into the expression: 2(1/4)(cos(pi/12))

5. Simplify: 2 * 1/4 = 1/2.

Therefore, the simplified expression is 1/2.