To solve this problem, we need to use the concept of braking distance and the equation for braking distance. Braking distance is the distance required for a vehicle to come to a complete stop after the brakes are applied. It depends on several factors including the initial speed of the vehicle, the coefficient of friction between the tires and the road, and the gradient of the road.

Let's break down the problem step by step:

1. Understanding the given information:
- The driver is traveling at a speed of 75 km/hr.
- The driver requires 10 m less to stop after applying brakes while traveling up the gradient, compared to traveling down the same gradient.
- The coefficient of friction given is f = 0.40.

2. Determining the braking distance equation:
The equation for braking distance is given by:
Braking distance = (initial speed^2) / (2 * acceleration)

The acceleration can be calculated using the equation:
Acceleration = f * g * cos(theta)
where g is the acceleration due to gravity (9.8 m/s^2) and theta is the angle of the gradient.

3. Calculating the braking distance while traveling up the gradient:
Let the braking distance while traveling up the gradient be D_up.
Using the given information, we have:
D_up = (75 km/hr)^2 / (2 * f * g * cos(theta))

4. Calculating the braking distance while traveling down the gradient:
Let the braking distance while traveling down the gradient be D_down.
Using the given information, we have:
D_down = (75 km/hr)^2 / (2 * f * g * cos(-theta))

Note: The cosine of the negative angle (-theta) is the same as the cosine of the positive angle (theta).

5. Applying the given condition:
According to the problem, the driver requires 10 m less to stop while traveling up the gradient compared to traveling down the same gradient.
Therefore, we have:
D_up = D_down - 10 m

6. Substituting the values and simplifying the equation:
Substituting the expressions for D_up and D_down from steps 3 and 4 into the equation from step 5, we get:
(75 km/hr)^2 / (2 * f * g * cos(theta)) = (75 km/hr)^2 / (2 * f * g * cos(-theta)) - 10 m

Simplifying further, we can cancel out common terms and rearrange the equation to isolate cos(theta):
cos(theta) = 1 / (1 + 10 / (75 km/hr)^2)

7. Converting the speed from km/hr to m/s:
To calculate the value of cos(theta), we need to convert the speed from km/hr to m/s.
1 km/hr = (1000 m / 3600 s) m/s
Therefore, 75 km/hr = (75 * 1000 m / 3600 s) m/s

8. Calculating the value of cos(theta):
Substituting the values into the equation from step 6, we get:
cos(theta) = 1 / (1 + 10 / ((75 * 1000 m / 3600 s)^2))

Simplifying further, we get:
cos(theta) =