What is the general solution for cot^2(x)csc^2(x)+2csc^2(x)-cot^2(x)=2 ?

The general solution for cot^2(x)csc^2(x)+2csc^2(x)-cot^2(x)=2 is x=(3pi)/2+2pin,x= pi/2+2pin

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cot^2(x)csc^2(x)+2csc^2(x)-cot^2(x)=2

Solution

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) - cot^{2} (x) = 2

Solution

x = \frac{3 π}{2} + 2 πn, x = \frac{π}{2} + 2 πn

Degrees

x = 27 0^{\circ} + 36 0^{\circ} n, x = 9 0^{\circ} + 36 0^{\circ} n

Solution steps

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) - cot^{2} (x) = 2

Subtract

2

from both sides

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) - cot^{2} (x) - 2 = 0

Factor

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) - cot^{2} (x) - 2 : (cot^{2} (x) + 2) (csc (x) + 1) (csc (x) - 1)

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) - cot^{2} (x) - 2

= (cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x)) + (- cot^{2} (x) - 2)

Factor out

- 1

from

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

- cot^{2} (x) - 2

Factor out common term

- 1

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

Factor out

csc^{2} (x)

from

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x) : csc^{2} (x) (cot^{2} (x) + 2)

cot^{2} (x) csc^{2} (x) + 2 csc^{2} (x)

Factor out common term

csc^{2} (x)

= csc^{2} (x) (cot^{2} (x) + 2)

= - (cot^{2} (x) + 2) + csc^{2} (x) (cot^{2} (x) + 2)

Factor out common term

cot^{2} (x) + 2

= (cot^{2} (x) + 2) (csc^{2} (x) - 1)

Factor

csc^{2} (x) - 1 : (csc (x) + 1) (csc (x) - 1)

csc^{2} (x) - 1

Rewrite

1

1^{2}

= csc^{2} (x) - 1^{2}

Apply Difference of Two Squares Formula:

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

csc^{2} (x) - 1^{2} = (csc (x) + 1) (csc (x) - 1)

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

= (cot^{2} (x) + 2) (csc (x) + 1) (csc (x) - 1)

(cot^{2} (x) + 2) (csc (x) + 1) (csc (x) - 1) = 0

Solving each part separately

cot^{2} (x) + 2 = 0 or csc (x) + 1 = 0 or csc (x) - 1 = 0

cot^{2} (x) + 2 = 0 :

No Solution

cot^{2} (x) + 2 = 0

Solve by substitution

cot^{2} (x) + 2 = 0

Let:

cot (x) = u

u^{2} + 2 = 0

u^{2} + 2 = 0 : u = 2 i, u = - 2 i

u^{2} + 2 = 0

Move

2

to the right side

u^{2} + 2 = 0

Subtract

2

from both sides

u^{2} + 2 - 2 = 0 - 2

Simplify

u^{2} = - 2

u^{2} = - 2

For

x^{2} = f (a)

the solutions are

x = f (a), - f (a)

u = - 2, u = - - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

Simplify

- - 2 : - 2 i

- - 2

Simplify

- 2 : 2 i

- 2

Apply radical rule:

- a = - 1 a

- 2 = - 1 2

= - 1 2

Apply imaginary number rule:

- 1 = i

= 2 i

= - 2 i

u = 2 i, u = - 2 i

Substitute back

u = cot (x)

cot (x) = 2 i, cot (x) = - 2 i

cot (x) = 2 i, cot (x) = - 2 i

cot (x) = 2 i :

No Solution

cot (x) = 2 i

No Solution

cot (x) = - 2 i :

No Solution

cot (x) = - 2 i

No Solution

Combine all the solutions

No Solution

csc (x) + 1 = 0 : x = \frac{3 π}{2} + 2 πn

csc (x) + 1 = 0

Move

1

to the right side

csc (x) + 1 = 0

Subtract

1

from 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{3 π}{2} + 2 πn

x = \frac{3 π}{2} + 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{3 π}{2} + 2 πn, x = \frac{π}{2} + 2 πn

Popular Examples

cos(t)= 3/5 ,0<t< pi/2

cos (t) = \frac{3}{5}, 0 < t < \frac{π}{2}

cos(x)=(-2)/2

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

solvefor y,-ln(cos(x))=-2cos(x)*sin(y)solve for

y, - ln (cos (x)) = - 2 cos (x) \cdot sin (y)

2sin(a)+csc(a)=3

2 sin (a) + csc (a) = 3

cos(x)=(-2)/7

cos (x) = \frac{- 2}{7}

Frequently Asked Questions (FAQ)

What is the general solution for cot^2(x)csc^2(x)+2csc^2(x)-cot^2(x)=2 ?
The general solution for cot^2(x)csc^2(x)+2csc^2(x)-cot^2(x)=2 is x=(3pi)/2+2pin,x= pi/2+2pin