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

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

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

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

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

Solution steps

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

Factor

sec (x) csc (x) + 2 csc (x) : csc (x) (sec (x) + 2)

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

Combine all the solutions

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

sin (x) + 3 cos (x) = 1

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

\frac{1}{2} = cos (x)

cos (θ) = \frac{5}{13}

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

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