Proof:

To prove the equation cos(π/3 x) = 1/2(cosx - √3sinx), we will use the trigonometric identity for the cosine of a sum of angles.

Identity:
cos(α + β) = cosα * cosβ - sinα * sinβ

Step 1:
Let's start by substituting α = x and β = π/3 in the cos(α + β) identity.

cos(x + π/3) = cos(x) * cos(π/3) - sin(x) * sin(π/3)

Step 2:
We know that cos(π/3) = 1/2 and sin(π/3) = √3/2. Substituting these values, we get:

cos(x + π/3) = cos(x) * 1/2 - sin(x) * (√3/2)

Simplifying further:

cos(x + π/3) = 1/2 * cos(x) - (√3/2) * sin(x)

Step 3:
Rearranging the terms, we have:

cos(x + π/3) = 1/2 * (cos(x) - √3 * sin(x))

Step 4:
Now, let's consider the left-hand side of the original equation, cos(π/3 x). Using the angle sum identity, we can rewrite it as:

cos(π/3 x) = cos(x + π/3)

Step 5:
Comparing the left-hand side of the original equation (cos(π/3 x)) with the right-hand side obtained in Step 3 (1/2 * (cos(x) - √3 * sin(x))), we can see that they are equal.

Conclusion:
Therefore, we have proved that cos(π/3 x) = 1/2(cosx - √3sinx) using the trigonometric identity for the cosine of a sum of angles.