What is the general solution for 2sec(x)csc(x)=4csc(x) ?

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

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2sec(x)csc(x)=4csc(x)

2 sec (x) csc (x) = 4 csc (x)

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

Degrees

x = 6 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

2 sec (x) csc (x) = 4 csc (x)

Subtract

4 csc (x)

from both sides

2 sec (x) csc (x) - 4 csc (x) = 0

Factor

2 sec (x) csc (x) - 4 csc (x) : 2 csc (x) (sec (x) - 2)

2 sec (x) csc (x) - 4 csc (x)

Rewrite

- 4

2 \cdot 2

= 2 sec (x) csc (x) + 2 \cdot 2 csc (x)

Factor out common term

2 csc (x)

= 2 csc (x) (sec (x) - 2)

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

Solving each part separately

csc (x) = 0 or sec (x) - 2 = 0

csc (x) = 0 :

No Solution

csc (x) = 0

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

No Solution

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

sec (x) - 2 = 0

Move

2

to the right side

sec (x) - 2 = 0

Add

2

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

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

Combine all the solutions

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

tan (2 x) + 2 cos (x) = 0

cos (3 x) = 2

- sin (2 x) + 2 cos (x) = 0

8 cos^{2} (x) - 4 = 0

tan (x) = - 5

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