How do you solve: Cos (x/2) - cosx=1?

1 Answer
Nghi N.
Mar 19, 2016

pi, (2pi)/3, (4pi/3)

Explanation:

Apply the trig identity: cos 2a = 2cos^2a - 1
cos x = 2cos^2 (x/2) - 1. We get:
cos (x/2) - 2cos^2 (x/2) + 1 = 1
cos (x/2)( 1 + 2cos (x/2)) = 0
a. cos x/2 = 0 --> 2 solutions:
x/2 = pi/2 --> x = pi
x/2 = 3pi/2 --> x = 3pi or x = pi.
b. 2cos (x/2) = - 1 --> cos (x/2) = - 1/2 -->
Trig table --> 2 solutions:
x/2 = (2pi)/3 --> x = (4pi)/3
x/2 = (4pi)/3 --> x = (8pi)/3, or x = (2pi)/3
Answers for (0, 2pi):
pi, (2pi)/3, (4pi)/3

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