What is the general solution for cot(x)sec(x)+cot(x)=0 ?

The general solution for cot(x)sec(x)+cot(x)=0 is No Solution for x\in\mathbb{R}

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cot(x)sec(x)+cot(x)=0

cot (x) sec (x) + cot (x) = 0

No Solution for x \in R

Solution steps

cot (x) sec (x) + cot (x) = 0

Factor

cot (x) sec (x) + cot (x) : cot (x) (sec (x) + 1)

cot (x) sec (x) + cot (x)

Factor out common term

cot (x)

= cot (x) (sec (x) + 1)

cot (x) (sec (x) + 1) = 0

Solving each part separately

cot (x) = 0 or sec (x) + 1 = 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

sec (x) + 1 = 0 : x = π + 2 πn

sec (x) + 1 = 0

Move

1

to the right side

sec (x) + 1 = 0

Subtract

1

from both sides

sec (x) + 1 - 1 = 0 - 1

Simplify

sec (x) = - 1

sec (x) = - 1

General solutions for

sec (x) = - 1

sec (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} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} 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} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

x = π + 2 πn

x = π + 2 πn

Combine all the solutions

x = \frac{π}{2} + πn, x = π + 2 πn

Since the equation is undefined for:

\frac{π}{2} + πn, π + 2 πn

No Solution for x \in R

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

cos (2 θ) - cos (θ) = 0

cot (x) + 3 = 0

4 sin^{2} (θ) + 4 = 5

sec^{2} (x) - 6 tan (x) = - 4

What is the general solution for cot(x)sec(x)+cot(x)=0 ?
The general solution for cot(x)sec(x)+cot(x)=0 is No Solution for x\in\mathbb{R}