cos (2x) = sec (x)?

Question

Solve for x. please show work..

Answer 1

cos(2x) = sec(x)
cos²(x) - sin²(x) = sec(x)
1 - (sin²(x) / cos²(x)) = sec(x) / cos²(x))
1 - tan²(x) = sec³(x)
1 - (sec²(x) -1)= sec³(x)
2 - sec²(x) = sec³(x)
sec³(x) + sec²(x) - 2 = 0

Let y = sec(x)

y³ + y² - 2 = 0
(y - 1)(y² - 2) = 0
so y = 1, √2, -√2
This means x = 0, π/4, 3π/4

Answer 2

Remember that sec(x) = 1/cos(x) so that
cos(2x) = 1/cos(x) or
cos(x)*cos(2x) = 1
and the only way that can happen is if both tterms are +1 or -1. cos(x) = +1 => x = 0
cos(x) = -1 => x = π, but then cos(2π) = +1 so that doesn't work and the answer is x = 0

HTH ☺


Doug