Use trigonometric identities to solve each equation in the interval [0, 2π). cos 2 x - sin 2 x = 1

Question

Use trigonometric identities to solve each equation in the interval [0, 2π).

cos ² x - sin ² x = 1

Answer 1

cos^2(x)- sin^2(x) = 1

cos(a+b) = cosacosb - sinasinb

cos2a = cos^2(a) - sin^2(a)

cos2x = cos^2(x) - sin^2(x) = 1

2x=arccos1 = 0 or 2pi or 4pi

x = 0 or pi (but not 4pi/2 = 2pi)

2pi is not in the interval [0,2pi) given the parentheses at the right end

Answer 2

Since cos²x + sin²x = 1, sin²x = 1 - cos²x.

So, if cos²x - sin²x = 1, then cos²x - (1 - cos²x) = 1

2cos²x = 2

cos²x = 1

So, cosx = 1 or -1.

x = 0, π