To solve this equation, we can treat it as a quadratic equation in terms of cos x:

cos^2 x + 13cos x - 14 = 0

Now, we can factor this quadratic equation:

(cos x + 14)(cos x - 1) = 0

Setting each factor to zero and solving for cos x, we get:

cos x + 14 = 0 or cos x - 1 = 0
cos x = -14 or cos x = 1

However, the cosine function only has values between -1 and 1, so the only valid solution is:

cos x = 1

Therefore, the solution to the equation cos^2 x + 13 cos x = 14 is x = 0.