5 cos π/3 cos 2π/3 sin (π-π/3)

To simplify the expression 5 cos(π/3) cos(2π/3) sin(π-π/3), we can use the trigonometric identities.

1. Recall the double angle identity for cosine: cos(2θ) = 2cos^2(θ) - 1.
2. Also, remember the difference of angles identity for sine: sin(θ - φ) = sin(θ)cos(φ) - cos(θ)sin(φ).

Now let's simplify the expression step by step:

1. cos(2π/3) = 2cos^2(π/3) - 1
(Using the double angle identity)

2. cos(2π/3) = 2(1/2)^2 - 1
(Substituting cos(π/3) = 1/2)

3. cos(2π/3) = 1/2 - 1
(Simplifying)

4. cos(2π/3) = -1/2

Now let's simplify the remaining part of the expression:

5. sin(π - π/3) = sin(2π/3)
(Using the difference of angles identity)

6. sin(2π/3) = sin(π/3)
(Sine function is periodic with period 2π)

7. sin(π/3) = √3/2
(Using the value of sin(π/3))

Finally, we can substitute the values back into the original expression:

5 cos(π/3) cos(2π/3) sin(π-π/3) = 5 * (1/2) * (-1/2) * (√3/2)
= -5/4 * √3/2
= -5√3/8

So, the simplified expression is -5√3/8.