Tính Sinx + cosx - 8sinx cosx = 0

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

First, let's rewrite the equation as sin(x) + cos(x) = 8sin(x)cos(x).

Next, we can use the identity sin(2x) = 2sin(x)cos(x) to rewrite the equation as sin(x) + cos(x) = 4sin(2x).

Now, let's square both sides of the equation to eliminate the square root: (sin(x) + cos(x))^2 = (4sin(2x))^2.

Expanding the left side of the equation, we get sin^2(x) + 2sin(x)cos(x) + cos^2(x) = 16sin^2(2x).

Using the identity sin^2(x) + cos^2(x) = 1, we can simplify the equation to 1 + 2sin(x)cos(x) = 16sin^2(2x).

Now, let's use the double angle identity sin(2x) = 2sin(x)cos(x) to rewrite the equation as 1 + sin(2x) = 16sin^2(2x).

Expanding the right side of the equation, we get 1 + sin(2x) = 16(1 - cos^2(2x)).

Using the identity sin^2(x) + cos^2(x) = 1, we can simplify the equation to 1 + sin(2x) = 16 - 16cos^2(2x).

Rearranging the equation, we get 16cos^2(2x) + sin(2x) = 15.

Now, we can use the double angle identity cos(2x) = cos^2(x) - sin^2(x) to rewrite the equation as 16(1 - sin^2(2x)) + sin(2x) = 15.

Expanding the left side of the equation, we get 16 - 16sin^2(2x) + sin(2x) = 15.

Rearranging the equation, we get 16sin^2(2x) - sin(2x) + 1 = 0.

Now, let's substitute u = sin(2x) to simplify the equation: 16u^2 - u + 1 = 0.

Using the quadratic formula, we can solve for u: u = (-(-1) ± √((-1)^2 - 4(16)(1))) / (2(16)).

Simplifying the equation, we get u = (1 ± √(1 - 64)) / 32.

Since the discriminant (1 - 64) is negative, there are no real solutions for u.

Therefore, there are no real solutions for the equation sin(x) + cos(x) - 8sin(x)cos(x) = 0.