What is the general solution for cos(2x)-cos(x)=1 ?

The general solution for cos(2x)-cos(x)=1 is x=2.46670…+2pin,x=-2.46670…+2pin

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cos(2x)-cos(x)=1

cos (2 x) - cos (x) = 1

x = 2.46670 \dots + 2 πn, x = - 2.46670 \dots + 2 πn

Solution steps

cos (2 x) - cos (x) = 1

Subtract

1

from both sides

cos (2 x) - cos (x) - 1 = 0

Rewrite using trig identities

- 2 - cos (x) + 2 cos^{2} (x) = 0

Solve by substitution

cos (x) = \frac{1 + 17}{4}, cos (x) = \frac{1 - 17}{4}

cos (x) = \frac{1 + 17}{4} :

No Solution

cos (x) = \frac{1 - 17}{4} : x = arccos (\frac{1 - 17}{4}) + 2 πn, x = - arccos (\frac{1 - 17}{4}) + 2 πn

Combine all the solutions

x = arccos (\frac{1 - 17}{4}) + 2 πn, x = - arccos (\frac{1 - 17}{4}) + 2 πn

Show solutions in decimal form

x = 2.46670 \dots + 2 πn, x = - 2.46670 \dots + 2 πn

1 - cot (x) = csc (x)

2 csc^{2} (x) = cot (x)

6 sin (x) cos (x) + 3 sin (x) - 2 cos (x) - 1 = 0

a cos (x) + b sin (x) = 0

cot (x) = \frac{7}{4}, sin (x) < 0

What is the general solution for cos(2x)-cos(x)=1 ?
The general solution for cos(2x)-cos(x)=1 is x=2.46670…+2pin,x=-2.46670…+2pin