Question

Find all solutions in the interval [0,2pi) for the equation cos2X - tanx = 1. Solutions must be in exact values.

Answer 1

Replace in the equation cos 2x = 1 - 2sin^2 x
(1 - 2si^2 x) - sin x/cos x = 1
2sin^2 x*cos x - sin x = 0
(sin x) (2sin x*cos x - 1) = 0
sin x*(sin 2x - 1) = 0. Solve the 2 basic trig equations.
a) sin x = 0 --> sin x = sin 0; or sin Pi, or sin 2Pi
Then x = ....., and x = .....; and x = .....
b) sin 2x = 1/2 --> sin 2x = Pi/6 --> 2x = Pi/6 --> x = .....
The trig unit circle gives another arc 2x that has the same sin value (1/2).
sin 2x = sin 5Pi/6 --> 2x = 5Pi/6 --> x = ......
Answers: total 5 values of x