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

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

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tan(x+pi)+cos(x+pi/2)=0

tan (x + π) + cos (x + \frac{π}{2}) = 0

x = 2 πn, x = π + 2 πn

Solution steps

tan (x + π) + cos (x + \frac{π}{2}) = 0

Rewrite using trig identities

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

Simplify

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

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

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

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

Factor

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

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cos (x) - 1 = 0 : x = 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

sin (θ) cos (θ) = \frac{3}{4}

tan (x + \frac{π}{6}) = 3

cos (2 θ) - sin^{2} (θ) = 0

\frac{sin ( 8 0 ^{\circ} )}{5} = \frac{sin ( x )}{4}

2 sin (2 x) + 1 = 0, 0 \leq x \leq 2 π

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