-5 sin2x-16(sinx-cosx)+8=0

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To solve the equation -5sin(2x) - 16(sin(x) - cos(x)) + 8 = 0, we can use trigonometric identities to simplify it.

Let's start by expanding -5sin(2x) using the double-angle identity: sin(2x) = 2sin(x)cos(x).

-5sin(2x) = -5(2sin(x)cos(x)) = -10sin(x)cos(x)

Next, expand -16(sin(x) - cos(x)):

-16sin(x) + 16cos(x)

Substitute these simplified expressions back into the original equation:

-10sin(x)cos(x) - 16sin(x) + 16cos(x) + 8 = 0

Now, we can rearrange the terms and factor out common factors:

-2sin(x)(5cos(x) + 8) + 16(cos(x) + 5) = 0

Now, we have factored the original equation. To solve for x, we need to set each factor to zero:

-2sin(x) = 0 or 5cos(x) + 8 = 0
sin(x) = 0 cos(x) = -8/5

The solutions for sin(x)= 0 are x = 0, π

The solutions for cos(x) = -8/5 are no real solutions, as the cosine values are limited to [-1,1]

Therefore, the solutions for the original equation are x = 0, π.