1) 2sin^2x + sin x - 3 = 0
This is a quadratic equation in terms of sin(x). Let's denote sin(x) as y for simplicity.
So, the equation becomes 2y^2 + y - 3 = 0
Solving this quadratic equation, we find:
y = 1, y = -1.5
Since sin(x) is between -1 and 1, the solutions are sin(x) = 1 and sin(x) = -1.5, which is not possible. So, there are no real solutions for this equation.

2) cos^2(pi-x) - sin((pi/2)-x) = 0
Using trigonometric identities, cos^2(pi-x) = sin^2x and sin((pi/2)-x) = cos(x)
So, the equation becomes sin^2x - cos(x) = 0
Now, using the identity sin^2x + cos^2x = 1, we get:
(1 - cos^2x) - cos(x) = 0
Expanding, we get: cos^2x + cos(x) - 1 = 0
This is a quadratic equation in terms of cos(x). Solving for cos(x), we find:
cos(x) = (-1 ± √5)/2

3) 3sin(x) + 2cos(x) = 0
This is a linear combination of sine and cosine. To solve this equation, we can divide by cos(x) to get it in terms of tan(x):
3tan(x) + 2 = 0
tan(x) = -2/3

4) 3sin(x) + 4cos(x) = 1
This equation is a linear combination of sine and cosine. Dividing by cos(x) to get it in terms of tan(x):
3tan(x) + 4 = 1
3tan(x) = -3
tan(x) = -1

5) tan(x) = 3ctg(x)
tan(x) = 3/cot(x)
tan(x) = 3tan(x)
This equation simplifies to:
2tan(x) = 0
tan(x) = 0

6) 3tan^2(x) - √3tan(x) = 0
Factoring out tan(x), we get:
tan(x)(3tan(x) - √3) = 0
This implies tan(x) = 0 or tan(x) = √3/3

7) sin(3x) = cos(5x)
Using the angle addition formula for sine, sin(3x) = sin(90 - 5x)
So, we have:
3x = 90 - 5x + 360k, where k is an integer
8x = 90 + 360k
x = 11.25 + 45k, where k is an integer.