SOLUTION: Solve: cos(x) cos (6x) =-1

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Question 1159951: Solve: cos(x) cos (6x) =-1
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

cos%28x%29+cos+%286x%29+=-1
-> you know that -1%3C=cos%28x%29%3C=1

case 1: when cos%28x%29=-1 and cos%286x%29+=1

cos%28x%29=-1
x=cos%5E-1%28-1%29
x=2pi%2An%2Bpi
and cos+%286x%29+=1
general solutions for cos+%286x+%29=1 is:
6x=2pi%2An
x=%282pi%2An%29%2F6+
x=%28pi%2An%29%2F3

check: if n=1
x=2pi%2Bpi=3pi... for cos%28x%29
x=pi%2F3 ... for cos+%286x+%29
cos%283pi%29%2Acos%286%28pi%2F3%29%29+=-1->true

case 2: when cos%28x%29=1 and cos%286x%29+=-1
cos%28x%29=1
x=cos%5E-1%281%29
x=2pi%2An
cos%286x%29+=-1
6x=2pi%2An%2Bpi
x=2pi%2An%2F6+%2Bpi%2F6
x=%28pi%2An%29%2F3%2Bpi%2F6

check: if n=1
x=2pi ...for cos%28x%29
x=pi%2F3%2Bpi%2F6=pi%2F2... for cos%286x%29
cos%282pi%29%2Acos%286%28pi%2F2%29%29+=-1->true

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.

HINT.  This equation can be satisfied IF and ONLY IF  one of the two factors in left side is equal to -1, 
while the other factor is equal to 1.


Answer.   x = pi = 180°.