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

The general solution for cot(x)sin(x)+cot(x)=0 is x= pi/2+pin

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cot(x)sin(x)+cot(x)=0

cot (x) sin (x) + cot (x) = 0

x = \frac{π}{2} + πn

Degrees

x = 9 0^{\circ} + 18 0^{\circ} n

Solution steps

cot (x) sin (x) + cot (x) = 0

Factor

cot (x) sin (x) + cot (x) : cot (x) (sin (x) + 1)

cot (x) sin (x) + cot (x)

Factor out common term

cot (x)

= cot (x) (sin (x) + 1)

cot (x) (sin (x) + 1) = 0

Solving each part separately

cot (x) = 0 or sin (x) + 1 = 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

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

sin (x) + 1 = 0

Move

1

to the right side

sin (x) + 1 = 0

Subtract

1

from both sides

sin (x) + 1 - 1 = 0 - 1

Simplify

sin (x) = - 1

sin (x) = - 1

General solutions for

sin (x) = - 1

sin (x)

periodicity table with

2 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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 2 πn

Combine all the solutions

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

Merge Overlapping Intervals

x = \frac{π}{2} + πn

4 cos (2 θ) = 4 sin (2 θ)

tan (x) = - 3, 0 \leq x \leq 2 π

5 tan^{2} (A) - 7 = 8

tan (x) = \frac{28}{16}

2 sin^{2} (x) = 3 - sin (x)

What is the general solution for cot(x)sin(x)+cot(x)=0 ?
The general solution for cot(x)sin(x)+cot(x)=0 is x= pi/2+pin