2cos^3(x)-cos(x)=0

Solution

2 cos^{3} (x) - cos (x) = 0

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Solution steps

2 cos^{3} (x) - cos (x) = 0

Solve by substitution

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

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

cos (x) = - \frac{2}{2} : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

cos (x) = \frac{2}{2} : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

- 2 cos (2 x) = 3

(tan (x) - 1) (2 sin (x) - 3) = 0

tan (θ) (1 + tan^{2} (θ)) = 0

tan (x) = 9

sin (2 x) = 0, 0 \leq x \leq 2 π