Trig. Equations Help?

Question

Hi, I needed help on two problems:
1) 3sec^2 x-2 sec x-1 = 0
2) 2csc^2x - 4 cscx= 0

Answer 1

1) 3sec^2 x-2 sec x-1 = 0 gactor
(3 sec x +1)(sec x -1)=0
3 sec x +1=0
3 sec x=-1
sec x=-1/3 but for all real x, sec x ≥1 or -1≥ sec x so this is not a real answer.

sec x-1=0
sec x=1
x=0, π, 2π etc

2) 2csc^2x - 4 cscx= 0 factor
2csc x(csc x-2)=0 since cscx≠0
csc x -2=0
csc x=2
sin x=1/2
x=π/6 or 5π/6

Answer 2

Always convert to basic functions, sin & cos. sec= 1/cos, so for 1)

3/cos^2 x -2/cos x-1=0

Multiply through by cos^2 x:

3 - 2*cos x - cos^2 x = 0;

make the substitution u = cosx

u^2 +2u - 3 = 0

Factor this (u+3)*(u-1) = 0 then u = -3 and u = 1

so cos x=-3 and cos x=1, x = 0 and arccos(-3)

csc=1/sin

2/sin^2 x - 4/sin x = 0

2 - 4sin x = 0

sin x = .5; x = 30º