What is the general solution for tan(θ)+cot(θ)= 4/(sqrt(3)) ?

The general solution for tan(θ)+cot(θ)= 4/(sqrt(3)) is θ= pi/6+pin,θ= pi/3+pin

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tan(θ)+cot(θ)= 4/(sqrt(3))

tan (θ) + cot (θ) = \frac{4}{3}

θ = \frac{π}{6} + πn, θ = \frac{π}{3} + πn

Solution steps

tan (θ) + cot (θ) = \frac{4}{3}

Subtract

\frac{4}{3}

from both sides

tan (θ) + cot (θ) - \frac{4}{3} = 0

Simplify

tan (θ) + cot (θ) - \frac{4}{3} : \frac{3 tan ( θ ) + 3 cot ( θ ) - 4}{3}

\frac{3 tan ( θ ) + 3 cot ( θ ) - 4}{3} = 0

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

3 tan (θ) + 3 cot (θ) - 4 = 0

Rewrite using trig identities

- 4 + \frac{3}{cot ( θ )} + cot (θ) 3 = 0

Solve by substitution

cot (θ) = 3, cot (θ) = \frac{3}{3}

cot (θ) = 3 : θ = \frac{π}{6} + πn

cot (θ) = \frac{3}{3} : θ = \frac{π}{3} + πn

Combine all the solutions

θ = \frac{π}{6} + πn, θ = \frac{π}{3} + πn

sec (x) + 4 cos (x) = 5

- 3 sec (x) = 2

3 cos (x) csc (x) = 23 cos (x)

cos (2 x) = \frac{9}{41}

sec (5 x) = \frac{2}{3}

What is the general solution for tan(θ)+cot(θ)= 4/(sqrt(3)) ?
The general solution for tan(θ)+cot(θ)= 4/(sqrt(3)) is θ= pi/6+pin,θ= pi/3+pin