Explanation:

To find the values of 1/a, 1/b, and 1/c, we need to first simplify the given equation and then solve for each variable individually.

Simplification:

We are given that:
costheta = b cos(theta 2pi/3) = c cos(theta 4pi/3)

Using the trigonometric identity cos(theta) = cos(2pi - theta), we can rewrite the equation as:
costheta = b cos(2pi - theta 2pi/3) = c cos(2pi - theta 4pi/3)

Now, let's simplify the equation further.

Simplifying cos(2pi - theta 2pi/3):
cos(2pi - theta 2pi/3) = cos(2pi - 2pi/3 * theta) = cos(2pi/3 * theta)

Simplifying cos(2pi - theta 4pi/3):
cos(2pi - theta 4pi/3) = cos(2pi - 4pi/3 * theta) = cos(4pi/3 * theta)

Updated Equation:
costheta = b cos(2pi/3 * theta) = c cos(4pi/3 * theta)

Now, we have the equation:
costheta = b cos(2pi/3 * theta) = c cos(4pi/3 * theta)

Solving for 1/a:
To solve for 1/a, we need to isolate a in the equation. Divide both sides of the equation by cos(theta):
1/a = cos(2pi/3 * theta) / cos(theta)

Solving for 1/b:
To solve for 1/b, we need to isolate b in the equation. Divide both sides of the equation by cos(2pi/3 * theta):
1/b = cos(4pi/3 * theta) / cos(2pi/3 * theta)

Solving for 1/c:
To solve for 1/c, we need to isolate c in the equation. Divide both sides of the equation by cos(4pi/3 * theta):
1/c = cos(2pi/3 * theta) / cos(4pi/3 * theta)

Conclusion:
The values of 1/a, 1/b, and 1/c are given by the following equations:
1/a = cos(2pi/3 * theta) / cos(theta)
1/b = cos(4pi/3 * theta) / cos(2pi/3 * theta)
1/c = cos(2pi/3 * theta) / cos(4pi/3 * theta)

These equations allow us to find the reciprocal values of a, b, and c based on the given trigonometric equation.