What is the general solution for cot(x)cos^2(x)=2cot(x),0<= x<= 2pi ?

Q: What is the general solution for cot(x)cos^2(x)=2cot(x),0<= x<= 2pi ?

The general solution for cot(x)cos^2(x)=2cot(x),0<= x<= 2pi is x= pi/2 ,x=(3pi)/2

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cot(x)cos^2(x)=2cot(x),0<= x<= 2pi

Solution

cot (x) cos^{2} (x) = 2 cot (x), 0 \leq x \leq 2 π

Solution

x = \frac{π}{2}, x = \frac{3 π}{2}

Degrees

x = 9 0^{\circ}, x = 27 0^{\circ}

Solution steps

cot (x) cos^{2} (x) = 2 cot (x), 0 \leq x \leq 2 π

Subtract

2 cot (x)

from both sides

cos^{2} (x) cot (x) - 2 cot (x) = 0

Factor

cos^{2} (x) cot (x) - 2 cot (x) : cot (x) (cos (x) + 2) (cos (x) - 2)

cos^{2} (x) cot (x) - 2 cot (x)

Factor out common term

cot (x)

= cot (x) (cos^{2} (x) - 2)

Factor

cos^{2} (x) - 2 : (cos (x) + 2) (cos (x) - 2)

cos^{2} (x) - 2

Apply radical rule:

a = (a)^{2}

2 = (2)^{2}

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

Apply Difference of Two Squares Formula:

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

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

= (cos (x) + 2) (cos (x) - 2)

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

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

Solving each part separately

cot (x) = 0 or cos (x) + 2 = 0 or cos (x) - 2 = 0

cot (x) = 0, 0 \leq x \leq 2 π : x = \frac{π}{2}, x = \frac{3 π}{2}

cot (x) = 0, 0 \leq x \leq 2 π

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

Solutions for the range

0 \leq x \leq 2 π

x = \frac{π}{2}, x = \frac{3 π}{2}

cos (x) + 2 = 0, 0 \leq x \leq 2 π :

No Solution

cos (x) + 2 = 0, 0 \leq x \leq 2 π

Move

2

to the right side

cos (x) + 2 = 0

Subtract

2

from both sides

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

Simplify

cos (x) = - 2

cos (x) = - 2

- 1 \leq cos (x) \leq 1

No Solution

cos (x) - 2 = 0, 0 \leq x \leq 2 π :

No Solution

cos (x) - 2 = 0, 0 \leq x \leq 2 π

Move

2

to the right side

cos (x) - 2 = 0

Add

2

to both sides

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

Simplify

cos (x) = 2

cos (x) = 2

- 1 \leq cos (x) \leq 1

No Solution

Combine all the solutions

x = \frac{π}{2}, x = \frac{3 π}{2}

Popular Examples

2cot^2(3θ)=5csc(3θ)-4

2 cot^{2} (3 θ) = 5 csc (3 θ) - 4

1-cos^2(x/2)=cos^2(x)

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

1=2sin(2θ)

1 = 2 sin (2 θ)

cos(-x)=(sqrt(3))/2

cos (- x) = \frac{3}{2}

sin(x)=0.61366

sin (x) = 0.61366

Frequently Asked Questions (FAQ)

What is the general solution for cot(x)cos^2(x)=2cot(x),0<= x<= 2pi ?
The general solution for cot(x)cos^2(x)=2cot(x),0<= x<= 2pi is x= pi/2 ,x=(3pi)/2