To solve the equation cos(2x)cos(5x) = cos(7x), we can use the trigonometric identity:

cos(A)cos(B) = (1/2)(cos(A-B) + cos(A+B))

Applying this identity to the equation, we have:

(1/2)(cos(2x-5x) + cos(2x+5x)) = cos(7x)

Simplifying the equation further:

(1/2)(cos(-3x) + cos(7x)) = cos(7x)

Now, let's isolate the cosine term:

(1/2)cos(-3x) + (1/2)cos(7x) = cos(7x)

Subtracting (1/2)cos(7x) from both sides:

(1/2)cos(-3x) = (1/2)cos(7x) - cos(7x)

Simplifying:

(1/2)cos(-3x) = -(1/2)cos(7x)

Now, we can use the fact that cos(-A) = cos(A) to simplify further:

(1/2)cos(3x) = -(1/2)cos(7x)

Multiplying both sides by 2:

cos(3x) = -cos(7x)

Now, we have a new equation to solve: cos(3x) = -cos(7x)

To solve this equation, we can use the fact that cos(A) = -cos(B) when A = B + π or A = -B + π.

So, we have two cases to consider:

Case 1: 3x = 7x + π

Simplifying:

4x = π

x = π/4

Case 2: 3x = -7x + π

Simplifying:

10x = π

x = π/10

Therefore, the solutions to the equation cos(2x)cos(5x) = cos(7x) are x = π/4 and x = π/10.