4cos^2(x)tan(x)-tan(x)=0

Solution

4 cos^{2} (x) tan (x) - tan (x) = 0

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

Solution steps

4 cos^{2} (x) tan (x) - tan (x) = 0

Factor

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

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

Solving each part separately

tan (x) = 0 or 2 cos (x) + 1 = 0 or 2 cos (x) - 1 = 0

tan (x) = 0 : x = πn

2 cos (x) + 1 = 0 : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

2 cos (x) - 1 = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

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

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

csc (x) = \frac{1}{cos ( x )}

csc (θ) = (1.201)

3 cos (θ) + 2 = 5 sec (θ), 0 \leq θ < 2 π

2 cos (\frac{π}{4} - 3 x) = 2