What is the general solution for -sec(θ)cot(θ)-sec(θ)=-2sec(θ) ?

The general solution for -sec(θ)cot(θ)-sec(θ)=-2sec(θ) is θ= pi/4+pin

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-sec(θ)cot(θ)-sec(θ)=-2sec(θ)

- sec (θ) cot (θ) - sec (θ) = - 2 sec (θ)

θ = \frac{π}{4} + πn

Degrees

θ = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

- sec (θ) cot (θ) - sec (θ) = - 2 sec (θ)

Subtract

- 2 sec (θ)

from both sides

- sec (θ) cot (θ) + sec (θ) = 0

Factor

- sec (θ) cot (θ) + sec (θ) : - sec (θ) (cot (θ) - 1)

- sec (θ) cot (θ) + sec (θ)

Factor out common term

- sec (θ)

= - sec (θ) (cot (θ) - 1)

- sec (θ) (cot (θ) - 1) = 0

Solving each part separately

sec (θ) = 0 or cot (θ) - 1 = 0

sec (θ) = 0 :

No Solution

sec (θ) = 0

sec (x) \leq - 1 or sec (x) \geq 1

No Solution

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

Combine all the solutions

θ = \frac{π}{4} + πn

sec^{2} (a) = \frac{4}{3}

sin (5 x - 27) = cos (6 x + 13.6)

sin (x) = \frac{1}{cos ( - 10 0 ^{\circ} )}

3 tan^{2} (x) - 23 tan (x) + 1 = 0

tan (θ) = 0.6795786612

What is the general solution for -sec(θ)cot(θ)-sec(θ)=-2sec(θ) ?
The general solution for -sec(θ)cot(θ)-sec(θ)=-2sec(θ) is θ= pi/4+pin