What is the general solution for tan(x+pi)+2sin(x+pi)=0 ?

The general solution for tan(x+pi)+2sin(x+pi)=0 is x=2pin,x=pi+2pin,x= pi/3+2pin,x=(5pi)/3+2pin

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tan(x+pi)+2sin(x+pi)=0

tan (x + π) + 2 sin (x + π) = 0

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

Solution steps

tan (x + π) + 2 sin (x + π) = 0

Rewrite using trig identities

\frac{sin ( x )}{cos ( x )} - 2 sin (x) = 0

Simplify

\frac{sin ( x )}{cos ( x )} - 2 sin (x) : \frac{sin ( x ) - 2 sin ( x ) cos ( x )}{cos ( x )}

\frac{sin ( x ) - 2 sin ( x ) cos ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Factor

sin (x) - 2 sin (x) cos (x) : - sin (x) (2 cos (x) - 1)

- sin (x) (2 cos (x) - 1) = 0

Solving each part separately

sin (x) = 0 or 2 cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

2 cos (x) - 1 = 0 : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

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

sin (x) = \frac{3}{2}

cos^{2} (x) = - cos (x)

sin (θ) - 1 = cos (θ)

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

3 sec (x) + 2 = 0

What is the general solution for tan(x+pi)+2sin(x+pi)=0 ?
The general solution for tan(x+pi)+2sin(x+pi)=0 is x=2pin,x=pi+2pin,x= pi/3+2pin,x=(5pi)/3+2pin