To solve this trigonometric equation, we can use the angle sum and difference identities for cosine and sine.

cos(40) = cos(20+20) = cos(20)cos(20) - sin(20)sin(20) = cos^2(20) - sin^2(20)

cos(20) = cos(10+10) = cos(10)cos(10) - sin(10)sin(10)

sin(10) = sin(20-10) = sin(20)cos(10) - cos(20)sin(10)

Now substitute these identities into the original equation:

cos(40) - cos(20) + sin(10) = 0
(cos^2(20) - sin^2(20)) - (cos(10)cos(10) - sin(10)sin(10)) + (sin(20)cos(10) - cos(20)sin(10)) = 0

Now, simplify the equation by substituting the identities above:

(cos^2(20) - sin^2(20)) - (cos^2(10) - sin^2(10)) + (sin(20)cos(10) - cos(20)sin(10)) = 0
(cos^2(20) - sin^2(20)) - (cos^2(10) - sin^2(10)) + (sin(20)cos(10) - cos(20)sin(10)) = 0

Since cos^2(20) - sin^2(20) = cos(40) and cos^2(10) - sin^2(10) = cos(20), the equation simplifies to:

cos(40) - cos(20) + sin(20)cos(10) - cos(20)sin(10) = 0

Therefore, the equation is:

cos(40) - cos(20) + sin(20)cos(10) - cos(20)sin(10) = 0

Unfortunately, it cannot be simplified further without an approximation or using trigonometric identities differently.