What is the general solution for 5sec(x)csc(x)=10csc(x) ?

The general solution for 5sec(x)csc(x)=10csc(x) is x= pi/3+2pin,x=(5pi)/3+2pin

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5sec(x)csc(x)=10csc(x)

5 sec (x) csc (x) = 10 csc (x)

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

Solution steps

5 sec (x) csc (x) = 10 csc (x)

Subtract

10 csc (x)

from both sides

5 sec (x) csc (x) - 10 csc (x) = 0

Factor

5 sec (x) csc (x) - 10 csc (x) : 5 csc (x) (sec (x) - 2)

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

Solving each part separately

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

csc (x) = 0 :

No Solution

sec (x) - 2 = 0 : 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

cos (x) = 1 - sin (x)

2 sin (2 θ) = cos (2 θ), 0 \leq θ \leq 2 π

tan (4 x) - 1 = 0

\frac{cos ( x ) cot ( x )}{1 - sin ( x )} = 3

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

What is the general solution for 5sec(x)csc(x)=10csc(x) ?
The general solution for 5sec(x)csc(x)=10csc(x) is x= pi/3+2pin,x=(5pi)/3+2pin