What is the general solution for cot(x)-cot(x)csc(x)=0 ?

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

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cot(x)-cot(x)csc(x)=0

cot (x) - cot (x) csc (x) = 0

x = \frac{π}{2} + πn

Degrees

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

Solution steps

cot (x) - cot (x) csc (x) = 0

Factor

cot (x) - cot (x) csc (x) : - cot (x) (csc (x) - 1)

cot (x) - cot (x) csc (x)

Factor out common term

- cot (x)

= - cot (x) (- 1 + csc (x))

- cot (x) (csc (x) - 1) = 0

Solving each part separately

cot (x) = 0 or csc (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

csc (x) - 1 = 0 : x = \frac{π}{2} + 2 πn

csc (x) - 1 = 0

Move

1

to the right side

csc (x) - 1 = 0

Add

1

to both sides

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

Simplify

csc (x) = 1

csc (x) = 1

General solutions for

csc (x) = 1

csc (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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 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} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{π}{2} + 2 πn

x = \frac{π}{2} + 2 πn

Combine all the solutions

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

Merge Overlapping Intervals

x = \frac{π}{2} + πn

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