Please help me, i need desprate help?

Question

I have a math test on thursday on identities and equations but i have trouble doing it, if you guys could help, really appreciate it, ill list some questions here and you can explain to me step by step how it works. thanks

Solve each quation algebraically for 0 < x < 2pie. Express roots in exact forms..
Cos(square)x - sin(square)x = 1
Sin(square)x - cos(square)x = 1

Consider the identity cos(pie - x) = -cosx
verify the identity numerically, when x = pie/6
for what values of x is the identity defined?

Answer 1

The first two problems are very similar. We use the identity:
(sinx)^2 + (cosx)^2 = 1
(sinx)^2 = 1 - (cosx)^2
Substituting into your first equation:
(cosx)^2 - (1 - (cosx)^2) = 1
2(cosx)^2 = 2
(cosx)^2 = 1
cosx = +/- 1
There's only one solution on the open interval (0, 2π)
x = π

(sinx)^2 - (cosx)^2 = 1
Again use (sinx)^2 + (cosx)^2 = 1
(cosx)^2 = 1 - (sinx)^2
Substitute into your second equation:
(sinx)^2 - (1 - (sinx)^2) = 1
2(sinx)^2 = 2
(sinx)^2 = 1
x = π/2, 3π/2

cos(π - x) = -cosx
cos(π - π/6) = -cos(π/6)
cos(5π/6) = -cos(π/6)
- (√3)/2 = - (√3)/2