1-3sinx *cos x + cos^2 x =0

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To solve this equation, we can simplify it first:

1 - 3sin(x) * cos(x) + cos^2(x) = 0

Now, let's rewrite cos^2(x) as 1 - sin^2(x):

1 - 3sin(x) * cos(x) + 1 - sin^2(x) = 0

2 - 3sin(x) * cos(x) - sin^2(x) = 0

Now, we can write sin^2(x) as (sin(x))^2:

2 - 3sin(x) * cos(x) - (sin(x))^2 = 0

Now, let's factor this equation:

(2 - sin(x))(1 + sin(x)) - 3sin(x) = 0

Now, we can solve for sin(x):

(2 - sin(x))(1 + sin(x)) - 3sin(x) = 0

Expanding this further:

2 + 2sin(x) - sin(x) - (sin(x))^2 - 3sin(x) = 0

2 - 2sin(x) - (sin(x))^2 = 0

Rearranging the terms:

-(sin(x))^2 - 2sin(x) + 2 = 0

Now, we have a quadratic equation in terms of sin(x):

sin^2(x) + 2sin(x) - 2 = 0

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

sin(x) = (-b ± sqrt(b^2 - 4ac)) / 2a

sin(x) = (-2 ± sqrt(2^2 - 4 1 -2)) / 2 * 1
sin(x) = (-2 ± sqrt(4 + 8)) / 2
sin(x) = (-2 ± sqrt(12)) / 2
sin(x) = (-2 ± 2sqrt(3)) / 2
sin(x) = -1 ± sqrt(3)

Therefore, the solutions for sin(x) are sin(x) = -1 + sqrt(3) and sin(x) = -1 - sqrt(3).