The first term \(a_1 = 1\) and the second term \(a_2 = \cos^2 x\). The common difference \(d\) is given by:

\(d = a_2 - a_1 = \cos^2 x - 1\).

The sum of the first ten terms \(S_{10}\) of an arithmetic progression is given by:

\(S_{10} = \frac{10}{2} (2a_1 + 9d)\).

Substitute \(a_1 = 1\) and \(d = \cos^2 x - 1\):

\(S_{10} = 5 [2 \times 1 + 9(\cos^2 x - 1)]\).

Simplify the expression:

\(S_{10} = 5 [2 + 9\cos^2 x - 9]\).

\(S_{10} = 5 [-7 + 9\cos^2 x]\).

Using the identity \(\cos^2 x = 1 - \sin^2 x\), substitute:

\(S_{10} = 5 [-7 + 9(1 - \sin^2 x)]\).

\(S_{10} = 5 [-7 + 9 - 9\sin^2 x]\).

\(S_{10} = 5 [2 - 9\sin^2 x]\).

\(S_{10} = 10 - 45\sin^2 x\).

Thus, \(a = 10\) and \(b = 45\).