To solve this equation, we can use the trigonometric identity that relates cosine and sine:

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

Substitute cos^2(x) with 1 - sin^2(x):

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

Rearrange the terms and get everything on one side:

sin^2(x) + 2sin(x) - 1/4 = 0

Now, we can solve this quadratic equation for sin(x):

Let's substitute sin(x) with y:

y^2 + 2y - 1/4 = 0

Now, solve the quadratic equation for y:

y = (-b ± sqrt(b^2 - 4ac))/(2a)

y = (-2 ± sqrt(2^2 - 41(-1/4)))/(2*1)
y = (-2 ± sqrt(4 + 1))/2
y = (-2 ± sqrt(5))/2

Now, substitute back sin(x) to y:

sin(x) = (-2 ± sqrt(5))/2

Therefore, the solutions for the equation are:

sin(x) = (-2 + √5)/2
sin(x) = (-2 - √5)/2