What is the general solution for (cot(θ)-1)(csc(θ)+1)=0 ?

The general solution for (cot(θ)-1)(csc(θ)+1)=0 is θ= pi/4+pin,θ=(3pi)/2+2pin

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(cot(θ)-1)(csc(θ)+1)=0

(cot (θ) - 1) (csc (θ) + 1) = 0

θ = \frac{π}{4} + πn, θ = \frac{3 π}{2} + 2 πn

Degrees

θ = 4 5^{\circ} + 18 0^{\circ} n, θ = 27 0^{\circ} + 36 0^{\circ} n

Solution steps

(cot (θ) - 1) (csc (θ) + 1) = 0

Solving each part separately

cot (θ) - 1 = 0 or csc (θ) + 1 = 0

cot (θ) - 1 = 0 : θ = \frac{π}{4} + πn

cot (θ) - 1 = 0

Move

1

to the right side

cot (θ) - 1 = 0

Add

1

to both sides

cot (θ) - 1 + 1 = 0 + 1

Simplify

cot (θ) = 1

cot (θ) = 1

General solutions for

cot (θ) = 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

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

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

csc (θ) + 1 = 0

Move

1

to the right side

csc (θ) + 1 = 0

Subtract

1

from both sides

csc (θ) + 1 - 1 = 0 - 1

Simplify

csc (θ) = - 1

csc (θ) = - 1

General solutions for

csc (θ) = - 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

θ = \frac{3 π}{2} + 2 πn

θ = \frac{3 π}{2} + 2 πn

Combine all the solutions

θ = \frac{π}{4} + πn, θ = \frac{3 π}{2} + 2 πn

tan (θ) = - \frac{2}{3}

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

sin (θ) = - \frac{2}{3}

50 sin^{2} (x) = 50 + 25 cos (x)

arctan (x) = π

What is the general solution for (cot(θ)-1)(csc(θ)+1)=0 ?
The general solution for (cot(θ)-1)(csc(θ)+1)=0 is θ= pi/4+pin,θ=(3pi)/2+2pin