What is the general solution for tan(x)cot(x)+2cot(x)=0 ?

The general solution for tan(x)cot(x)+2cot(x)=0 is x=-1.10714…+pin

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tan(x)cot(x)+2cot(x)=0

tan (x) cot (x) + 2 cot (x) = 0

x = - 1.10714 \dots + πn

Degrees

x = - 63.43494 \dots^{\circ} + 18 0^{\circ} n

Solution steps

tan (x) cot (x) + 2 cot (x) = 0

Factor

tan (x) cot (x) + 2 cot (x) : cot (x) (tan (x) + 2)

tan (x) cot (x) + 2 cot (x)

Factor out common term

cot (x)

= cot (x) (tan (x) + 2)

cot (x) (tan (x) + 2) = 0

Solving each part separately

cot (x) = 0 or tan (x) + 2 = 0

cot (x) = 0 : x = \frac{π}{2} + πn

cot (x) = 0

General solutions for

cot (x) = 0

cot (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

tan (x) + 2 = 0 : x = arctan (- 2) + πn

tan (x) + 2 = 0

Move

2

to the right side

tan (x) + 2 = 0

Subtract

2

from both sides

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

Simplify

tan (x) = - 2

tan (x) = - 2

Apply trig inverse properties

tan (x) = - 2

General solutions for

tan (x) = - 2

tan (x) = - a \Rightarrow x = arctan (- a) + πn

x = arctan (- 2) + πn

x = arctan (- 2) + πn

Combine all the solutions

x = \frac{π}{2} + πn, x = arctan (- 2) + πn

Since the equation is undefined for:

\frac{π}{2} + πn

x = arctan (- 2) + πn

Show solutions in decimal form

x = - 1.10714 \dots + πn

What is the general solution for tan(x)cot(x)+2cot(x)=0 ?
The general solution for tan(x)cot(x)+2cot(x)=0 is x=-1.10714…+pin