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

Solution

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

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

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

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Factor

- 2 cos (2 x) cos (x) + 2 cos (x) sin (x) : 2 cos (x) (- cos (2 x) + sin (x))

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

Solving each part separately

cos (x) = 0 or - cos (2 x) + sin (x) = 0

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

- cos (2 x) + sin (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

5 sin^{2} (θ) = 2 sin (θ)

1 + cos (x) = 0

tan^{2} (θ) = \frac{1}{3}

3 cot (x) - 1 = 0

cos (x) + 1 = - cos (x)