What is the general solution for cot(x)sin^2(x)=4cot(x) ?

The general solution for cot(x)sin^2(x)=4cot(x) is x= pi/2+pin

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cot(x)sin^2(x)=4cot(x)

Solution

cot (x) sin^{2} (x) = 4 cot (x)

Solution

x = \frac{π}{2} + πn

Degrees

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

Solution steps

cot (x) sin^{2} (x) = 4 cot (x)

Subtract

4 cot (x)

from both sides

sin^{2} (x) cot (x) - 4 cot (x) = 0

Factor

sin^{2} (x) cot (x) - 4 cot (x) : cot (x) (sin (x) + 2) (sin (x) - 2)

sin^{2} (x) cot (x) - 4 cot (x)

Factor out common term

cot (x)

= cot (x) (sin^{2} (x) - 4)

Factor

sin^{2} (x) - 4 : (sin (x) + 2) (sin (x) - 2)

sin^{2} (x) - 4

Rewrite

4

2^{2}

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

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (sin (x) + 2) (sin (x) - 2)

= cot (x) (sin (x) + 2) (sin (x) - 2)

cot (x) (sin (x) + 2) (sin (x) - 2) = 0

Solving each part separately

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

sin (x) + 2 = 0 :

No Solution

sin (x) + 2 = 0

Move

2

to the right side

sin (x) + 2 = 0

Subtract

2

from both sides

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

Simplify

sin (x) = - 2

sin (x) = - 2

- 1 \leq sin (x) \leq 1

No Solution

sin (x) - 2 = 0 :

No Solution

sin (x) - 2 = 0

Move

2

to the right side

sin (x) - 2 = 0

Add

2

to both sides

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

Simplify

sin (x) = 2

sin (x) = 2

- 1 \leq sin (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{2} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for cot(x)sin^2(x)=4cot(x) ?
The general solution for cot(x)sin^2(x)=4cot(x) is x= pi/2+pin