What is the general solution for tan(x)sec(x)=-2tan(x) ?

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

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tan(x)sec(x)=-2tan(x)

tan (x) sec (x) = - 2 tan (x)

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

Solution steps

tan (x) sec (x) = - 2 tan (x)

Subtract

- 2 tan (x)

from both sides

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

Factor

tan (x) sec (x) + 2 tan (x) : tan (x) (sec (x) + 2)

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

Solving each part separately

tan (x) = 0 or sec (x) + 2 = 0

tan (x) = 0 : x = πn

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

Combine all the solutions

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

1 + sin^{2} (x) = 2

sin (\frac{3 π}{2} + x) + sin (\frac{3 π}{2} + x) = - 2

cos (x) tan (x) = 0

cos^{2} (x) + 4 cos (x) + 3 = 0

sin (x) + sin^{2} (x) = 0

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