Tính nghiệm của phương trình

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

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 use another identity sin(x) = cos(π/2 - x) to rewrite the equation as cos(π/2 - x) + cos(x) = 4sin(2x).

Using the identity cos(a) + cos(b) = 2cos((a+b)/2)cos((a-b)/2), we can rewrite the equation as 2cos((π/2 - x + x)/2)cos((π/2 - x - x)/2) = 4sin(2x).

Simplifying further, we have 2cos(π/4)cos(-x/2) = 4sin(2x).

Using the identity cos(π/4) = sin(π/4) = 1/√2, we have 2(1/√2)cos(-x/2) = 4sin(2x).

Simplifying, we have √2cos(-x/2) = 2sin(2x).

Using the identity cos(-a) = cos(a), we have √2cos(x/2) = 2sin(2x).

Dividing both sides by 2, we have √2/2cos(x/2) = sin(2x).

Using the identity sin(2a) = 2sin(a)cos(a), we have √2/2cos(x/2) = 2sin(x)cos(x).

Dividing both sides by cos(x), we have √2/2cot(x/2) = 2sin(x).

Using the identity cot(a) = 1/tan(a), we have √2/2tan(x/2) = 2sin(x).

Multiplying both sides by 2, we have √2tan(x/2) = 4sin(x).

Using the identity tan(a/2) = sin(a)/(1+cos(a)), we have √2sin(x)/(1+cos(x)) = 4sin(x).

Multiplying both sides by (1+cos(x)), we have √2sin(x) = 4sin(x)(1+cos(x)).

Expanding the right side, we have √2sin(x) = 4sin(x) + 4sin(x)cos(x).

Subtracting 4sin(x) from both sides, we have √2sin(x) - 4sin(x) = 4sin(x)cos(x).

Combining like terms, we have (-3√2)sin(x) = 4sin(x)cos(x).

Dividing both sides by sin(x), we have -3√2 = 4cos(x).

Dividing both sides by 4, we have -3√2/4 = cos(x).

Simplifying, we have -√2/2 = cos(x).

Therefore, the solution to the equation sin(x) + cos(x) - 8sin(x)cos(x) = 0 is x = π/4 + 2nπ, where n is an integer.