the integral of sin^2 4θ dθ from -pi/4 to pi/4
There is a distinction between knowing the decimal expansion of a number, and its exact value. For example, your integral does have the value 0.785398, but that doesn't really mean anything mathematically, because its an approximation. The exact value in this case is actually $\displaystyle \frac{{\pi}}{{4}}$. You need to get the exact value.
One way to do that is using the identity I copied above, and then use the anti-derivative with the fundamental theorem of calculus to find the definite integral. You can use u-substitution to find the anti derivative, but it can be done otherwise:
$\displaystyle \sin^2(4\theta) = \frac{1 - \cos(8\theta)}{2}$
So
$\displaystyle \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} \frac{1 - \cos(8\theta)}{2} d\theta = \frac{\theta}{2} - \frac{\sin(8\theta)}{16} $ evaluated from -Pi/4 to Pi/4
$\displaystyle = \frac{\pi}{8} - \frac{\sin(2\pi)}{16} - (-\frac{\pi}{8}) + \frac{\sin(2\pi)}{16} = \frac{{\pi}}{{4}}$