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

The general solution for cos(2x)=2cos(x) is x=1.94553…+2pin,x=-1.94553…+2pin

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cos(2x)=2cos(x)

cos (2 x) = 2 cos (x)

x = 1.94553 \dots + 2 πn, x = - 1.94553 \dots + 2 πn

Solution steps

cos (2 x) = 2 cos (x)

Subtract

2 cos (x)

from both sides

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

Rewrite using trig identities

- 1 - 2 cos (x) + 2 cos^{2} (x) = 0

Solve by substitution

cos (x) = \frac{1 + 3}{2}, cos (x) = \frac{1 - 3}{2}

cos (x) = \frac{1 + 3}{2} :

No Solution

cos (x) = \frac{1 - 3}{2} : x = arccos (\frac{1 - 3}{2}) + 2 πn, x = - arccos (\frac{1 - 3}{2}) + 2 πn

Combine all the solutions

x = arccos (\frac{1 - 3}{2}) + 2 πn, x = - arccos (\frac{1 - 3}{2}) + 2 πn

Show solutions in decimal form

x = 1.94553 \dots + 2 πn, x = - 1.94553 \dots + 2 πn

sin (θ) + 3 cos (θ) = 1

6 tan^{2} (x) - 2 = 4

cos (2 x + \frac{π}{3}) = \frac{1}{2}

x, tan (x \cdot y!) = 1

sec^{2} (x) - 5 sec (x) = 0

What is the general solution for cos(2x)=2cos(x) ?
The general solution for cos(2x)=2cos(x) is x=1.94553…+2pin,x=-1.94553…+2pin