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

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

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

Solution

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

Solution

x = \frac{π}{2} + πn

Degrees

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

Solution steps

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

Subtract

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

from both sides

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

Factor

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

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

Factor out common term

- cot (x)

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

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

Popular Examples

arctan(x)=-1

arctan (x) = - 1

9cos(x)tan(x)=-4tan(x)

9 cos (x) tan (x) = - 4 tan (x)

cos(θ)-sin(θ)=0

cos (θ) - sin (θ) = 0

tan^2(x)-9tan(x)-10=0

tan^{2} (x) - 9 tan (x) - 10 = 0

(cot(θ)-1)(csc(θ)-1)=0

(cot (θ) - 1) (csc (θ) - 1) = 0

Frequently Asked Questions (FAQ)

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