What is the general solution for tan(5x)=cot(x) ?

The general solution for tan(5x)=cot(x) is x= pi/(12)+(pin)/3 ,x= pi/4+(pin)/3

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tan(5x)=cot(x)

tan (5 x) = cot (x)

x = \frac{π}{12} + \frac{πn}{3}, x = \frac{π}{4} + \frac{πn}{3}

Solution steps

tan (5 x) = cot (x)

Subtract

cot (x)

from both sides

tan (5 x) - cot (x) = 0

Express with sin, cos

\frac{- cos ( 5 x ) cos ( x ) + sin ( 5 x ) sin ( x )}{cos ( 5 x ) sin ( x )} = 0

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

- cos (5 x) cos (x) + sin (5 x) sin (x) = 0

Rewrite using trig identities

- cos (5 x + x) = 0

Divide both sides by

- 1

cos (5 x + x) = 0

General solutions for

cos (5 x + x) = 0

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

Solve

5 x + x = \frac{π}{2} + 2 πn : x = \frac{π}{12} + \frac{πn}{3}

Solve

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

x = \frac{π}{12} + \frac{πn}{3}, x = \frac{π}{4} + \frac{πn}{3}

\frac{1}{4} sin (8 x) = - \frac{1}{8}, 2 cos (8 x) = 3

sin (x) - sin (2 x) = 0, 0 \leq x < 2 π

sin (2 x) = \frac{- 2 6}{7}

arctan (4 x) + arctan (6 x) = 90

2 sin (θ) tan (θ) - tan (θ) = 1 - 2 sin (θ)

What is the general solution for tan(5x)=cot(x) ?
The general solution for tan(5x)=cot(x) is x= pi/(12)+(pin)/3 ,x= pi/4+(pin)/3