Question

Find x where pi <= x <= 2pi? 4 cos^2 x - 2= sec^2 x - tan^2 x

Answer 1

`(4pi)/3; (11pi)/6`

Explanation:

`4cos^2 x - 2 = sec^2 x - tan^2 x` (1)
Develop the right side:
`RS = 1/(cos^2 x) - sin^2 x/(cos^2 x) = (1 - sin^2 x)/(cos^2 x) = cos^2 x/(cos^2 x) = 1`
The equation (1) becomes:
`4cos^2 x - 2 = 1`
`cos^2 x = 3/4`
`cos x = +- sqrt3/2`
a. `cos x = sqrt3/2`
Trig table and unit circle give 2 solutions of x -->
`x = +- pi/6`
b. `cos x = - sqrt3/2` -->
`x = +- (2pi)/3`
Inside the interval `(pi, 2pi)`, the answers are:
`x = - pi/6`, or `x = (11pi)/6` (co-terminal)
`x = - (2pi)/3`, or `x = pi + pi/3 = (4pi)/3` (co-terminal)