To solve this equation, we can use trigonometric identities. First, let's simplify the equation:

7/4cos(x/4) = cos^3(x/4) + sin(x/2)

Since cos^3(x/4) is equal to (cos(x/4))^3, we can rewrite the equation as:

7/4cos(x/4) = (cos(x/4))^3 + sin(x/2)

Now, let's use the trigonometric identity: cos(2θ) = 1 - 2sin^2(θ)

By substituting θ = x/4, we get:

cos(x/2) = 1 - 2sin^2(x/4)

Rearranging the equation:

2sin^2(x/4) = 1 - cos(x/2)

Now, we substitute this back into our original equation:

7/4cos(x/4) = cos(x/4) * (1 - cos(x/2)) + sin(x/2)

Expanding the equation:

7/4cos(x/4) = cos(x/4) - cos(x/2)cos(x/4) + sin(x/2)

Rearrange the equation:

7/4cos(x/4) = cos(x/4) + sin(x/2) - (cos(x/2)sin(x/4))

Now, we can use the double angle identities:

sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos^2(θ) - sin^2(θ)

Using these identities, we can simplify the equation further. Feel free to simplify the equation and solve for x.