6sec^2(θ)tan(θ)=12tan(θ)

Solution

6 sec^{2} (θ) tan (θ) = 12 tan (θ)

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

Solution steps

6 sec^{2} (θ) tan (θ) = 12 tan (θ)

Subtract

12 tan (θ)

from both sides

6 sec^{2} (θ) tan (θ) - 12 tan (θ) = 0

Factor

6 sec^{2} (θ) tan (θ) - 12 tan (θ) : 6 tan (θ) (sec (θ) + 2) (sec (θ) - 2)

6 tan (θ) (sec (θ) + 2) (sec (θ) - 2) = 0

Solving each part separately

tan (θ) = 0 or sec (θ) + 2 = 0 or sec (θ) - 2 = 0

tan (θ) = 0 : θ = πn

sec (θ) + 2 = 0 : θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

sec (θ) - 2 = 0 : θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

Combine all the solutions

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

7 tan (x) = 33 + tan (x)

cos (2 x) = tan (2 x)

sin (\frac{θ}{2}) - 1 = 0

2 sin (4 x) + 3 = 0

7 sec (x) + 7 tan (x) = 7