5(1 - sin(2x)) - 16 (sin(x) - cos(x)) + 3 = 0

To solve the equation

\[
5(1 - \sin(2x)) - 16(\sin(x) - \cos(x)) + 3 = 0,
\]

let's start by simplifying the expression step by step:

1. **Expand the equation**:
\[
5 - 5\sin(2x) - 16\sin(x) + 16\cos(x) + 3 = 0.
\]

2. **Combine like terms**:
\[
8 - 5\sin(2x) - 16\sin(x) + 16\cos(x) = 0.
\]

3. **Use the double angle identity** for \(\sin(2x)\):
\[
\sin(2x) = 2\sin(x)\cos(x),
\]
thus,
\[
-5\sin(2x) = -10\sin(x)\cos(x).
\]

Substituting this back into the equation gives:
\[
8 - 10\sin(x)\cos(x) - 16\sin(x) + 16\cos(x) = 0.
\]

4. **Rearranging the terms**:
\[
-10\sin(x)\cos(x) - 16\sin(x) + 16\cos(x) + 8 = 0.
\]

5. **Factoring the expression**:
This equation can be approached by introducing new variables or using numerical methods since it may not factor easily, but let's convert terms and isolate a trigonometric function if possible.

6. **Isolate one of the trigonometric functions** (optional step - this depends on your preferred solving method):
For example: grouping terms in a manageable way may lead to solvable algebraic forms or trig identities.

At this point, you have a trigonometric equation that can be solved using numerical methods or further algebraic manipulation depending on the desired range of solutions.

If a numerical solution is required or if you need to solve this for specific intervals, feel free to let me know!