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

The general solution for (cos(2x))/(cos(x))=(cos(x))/(cos(x)) is x=(4pin)/3 ,x=(2pi)/3+(4pin)/3

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(cos(2x))/(cos(x))=(cos(x))/(cos(x))

\frac{cos ( 2 x )}{cos ( x )} = \frac{cos ( x )}{cos ( x )}

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

Solution steps

\frac{cos ( 2 x )}{cos ( x )} = \frac{cos ( x )}{cos ( x )}

Subtract

\frac{cos ( x )}{cos ( x )}

from both sides

\frac{cos ( 2 x )}{cos ( x )} - 1 = 0

Simplify

\frac{cos ( 2 x )}{cos ( x )} - 1 : \frac{cos ( 2 x ) - cos ( x )}{cos ( x )}

\frac{cos ( 2 x ) - cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Rewrite using trig identities

- 2 sin (\frac{3 x}{2}) sin (\frac{x}{2}) = 0

Solving each part separately

sin (\frac{3 x}{2}) = 0 or sin (\frac{x}{2}) = 0

sin (\frac{3 x}{2}) = 0 : x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

sin (\frac{x}{2}) = 0 : x = 4 πn, x = 2 π + 4 πn

Combine all the solutions

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}, x = 4 πn, x = 2 π + 4 πn

Merge Overlapping Intervals

x = \frac{4 πn}{3}, x = \frac{2 π}{3} + \frac{4 πn}{3}

tan^{2} (4 5^{\circ} + \frac{x}{3}) = 0.58

1 = 4 cos^{2} (x) - 4 cos (x) + 2

1 = cos (3 θ)

3 sin (2 x) + cos (x) = 0

sinh (a x) = 0

What is the general solution for (cos(2x))/(cos(x))=(cos(x))/(cos(x)) ?
The general solution for (cos(2x))/(cos(x))=(cos(x))/(cos(x)) is x=(4pin)/3 ,x=(2pi)/3+(4pin)/3