C

To find the maximum height attained by a projectile fired at an angle \( \theta \) with the vertical with an initial velocity \( u \), we use the formula for the vertical component of the velocity and the kinematic equations.
Step 1: Decompose the initial velocity into vertical and horizontal components. The vertical component is given by \( u_v = u \cos(\theta) \).
Step 2: The maximum height \( H \) is achieved when the vertical velocity becomes zero. Using the kinematic equation \( v^2 = u^2 + 2as \), where \( v = 0 \), \( a = -g \) (acceleration due to gravity), we get:
0 = \( (u \cos(\theta))^2 - 2gH \).
Step 3: Rearranging gives us: \( H = \frac{(u \cos(\theta))^2}{2g} \).
Hence, the answer matches Option C.