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

The general solution for (csc(x))=-csc(x)cot(x) is x=(3pi)/4+pin

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

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

x = \frac{3 π}{4} + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n

Solution steps

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

Subtract

- csc (x) cot (x)

from both sides

csc (x) + csc (x) cot (x) = 0

Factor

csc (x) + csc (x) cot (x) : csc (x) (cot (x) + 1)

csc (x) + csc (x) cot (x)

Factor out common term

csc (x)

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

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

Solving each part separately

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

csc (x) = 0 :

No Solution

csc (x) = 0

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

cot (x) + 1 = 0 : x = \frac{3 π}{4} + πn

cot (x) + 1 = 0

Move

1

to the right side

cot (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

cot (x) = - 1

cot (x) = - 1

General solutions for

cot (x) = - 1

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{3 π}{4} + πn

x = \frac{3 π}{4} + πn

Combine all the solutions

x = \frac{3 π}{4} + πn

cot (x) = - \frac{4}{3}

\frac{cos ^{2} ( x )}{sin ^{2} ( x )} = 1

x, f = \frac{cos ( 2 x )}{cos ( x ) - sin ( x )}

cos (2 x) + 5 = 6.25

sin (x) = 4 sin (8)

What is the general solution for (csc(x))=-csc(x)cot(x) ?
The general solution for (csc(x))=-csc(x)cot(x) is x=(3pi)/4+pin