What is the general solution for 2tan(x)csc(x)+2csc(x)+tan(x)=-1 ?

The general solution for 2tan(x)csc(x)+2csc(x)+tan(x)=-1 is x=(3pi)/4+pin

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2tan(x)csc(x)+2csc(x)+tan(x)=-1

2 tan (x) csc (x) + 2 csc (x) + tan (x) = - 1

x = \frac{3 π}{4} + πn

Solution steps

2 tan (x) csc (x) + 2 csc (x) + tan (x) = - 1

Subtract

- 1

from both sides

2 tan (x) csc (x) + 2 csc (x) + tan (x) + 1 = 0

Factor

2 tan (x) csc (x) + 2 csc (x) + tan (x) + 1 : (tan (x) + 1) (2 csc (x) + 1)

(tan (x) + 1) (2 csc (x) + 1) = 0

Solving each part separately

tan (x) + 1 = 0 or 2 csc (x) + 1 = 0

tan (x) + 1 = 0 : x = \frac{3 π}{4} + πn

2 csc (x) + 1 = 0 :

No Solution

Combine all the solutions

x = \frac{3 π}{4} + πn

4 cos (x) + 1 = 2 cos (x)

cos (2 x) + cos (4 x) + cos (6 x) = 0

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

6 sin (x) = 6 sin (2 x)

sin (5 x) + sin (x) = 3 cos (2 x)

What is the general solution for 2tan(x)csc(x)+2csc(x)+tan(x)=-1 ?
The general solution for 2tan(x)csc(x)+2csc(x)+tan(x)=-1 is x=(3pi)/4+pin