Solve algebraically? cos(x-Pi/4)+cos(x+pi/4)=1 for 0 ≤ x≤ 2pi

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Answer 1 · 2018-05-08T22:17:04

#x = pi/4 or x = {7pi}/4 #

#cos(x-pi/4) + cos(x+pi/4)=1#

We'll expand with the difference and sum angle formulas and see where we are.

#cos x cos (pi/4) + sin x sin(pi/4) + cos x cos(pi/4) - sin x sin(pi/4) = 1#

#2 cos x cos(pi/4) = 1 #

# 2 cos x (sqrt{2}/2) = 1#

#cos x = 1/sqrt{2} #

That's 45/45/90 in the first and fourth quadrant,

#x = pi/4 or x = {7pi}/4 #

Check:

#cos 0 + cos(pi/2)=1+0=1 quad sqrt #

#cos({6pi}/4) + cos({8pi}/4) = 0+1=1 quad sqrt #