How do you do these trig problems? solving for x

21. sin^2x= 3cos^2x

33. cscx + cotx= 1

35. cos x/2= sqrt2/2

For number one you can divide both sides by cos^2x and remembering that sinx / cosx = tanx,

tan^2x = 3
tanx = sqrt(3) (taking the square root of both sides)
x = arctan(sqrt 3)
x = pi / 3 radians

For number 2 it is easiest to put csc and cot in terms of sine and cosine. Remember that cscx = 1/sinx and cotx = cosx / sinx.

1/sinx + cosx/sinx = 1
1 + cosx = sinx (multiplying by sinx)

Now we need to square both sides
(1 + cosx)^2 = sin^2x
1 + 2cosx + cos^2x = sin^2x
1 + 2cosx + cos^2x = 1 - cos^2x
2 cosx + 2cos^2x = 0
cosx(1 + cosx) = 0
so cosx must either equal 0 or -1
This happens at pi / 2 and pi radians respectively.

For number 3:

x/2 = arccos(sqrt2/2)
x/2 = pi / 4 rad
x = pi / 2 rad

Hope I have been of some help :)