What is the general solution for cot(x)sec(x)+sqrt(2)cot(x)=0 ?

The general solution for cot(x)sec(x)+sqrt(2)cot(x)=0 is x=(3pi)/4+2pin,x=(5pi)/4+2pin

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cot(x)sec(x)+sqrt(2)cot(x)=0

cot (x) sec (x) + 2 cot (x) = 0

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Degrees

x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n

Solution steps

cot (x) sec (x) + 2 cot (x) = 0

Factor

cot (x) sec (x) + 2 cot (x) : cot (x) (sec (x) + 2)

cot (x) sec (x) + 2 cot (x)

Factor out common term

cot (x)

= cot (x) (sec (x) + 2)

cot (x) (sec (x) + 2) = 0

Solving each part separately

cot (x) = 0 or sec (x) + 2 = 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) + 2 = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) + 2 = 0

Move

2

to the right side

sec (x) + 2 = 0

Subtract

2

from both sides

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

Simplify

sec (x) = - 2

sec (x) = - 2

General solutions for

sec (x) = - 2

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 = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Combine all the solutions

x = \frac{π}{2} + πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

Since the equation is undefined for:

\frac{π}{2} + πn

x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

5 sin (x) - 20 = - 15

cos (x) = \frac{11}{14}

2 sin (θ) cos (θ) + 2 sin (θ) = 0

cot (3 8^{\circ}) = tan (2 x)

sin (x \cdot 240) = 0

What is the general solution for cot(x)sec(x)+sqrt(2)cot(x)=0 ?
The general solution for cot(x)sec(x)+sqrt(2)cot(x)=0 is x=(3pi)/4+2pin,x=(5pi)/4+2pin