What is the general solution for (sin(x))/(1+cos(x))+cot(x)=2 ?

The general solution for (sin(x))/(1+cos(x))+cot(x)=2 is x= pi/6+2pin,x=(5pi)/6+2pin

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(sin(x))/(1+cos(x))+cot(x)=2

\frac{sin ( x )}{1 + cos ( x )} + cot (x) = 2

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

\frac{sin ( x )}{1 + cos ( x )} + cot (x) = 2

Subtract

2

from both sides

\frac{sin ( x )}{1 + cos ( x )} + cot (x) - 2 = 0

Simplify

\frac{sin ( x )}{1 + cos ( x )} + cot (x) - 2 : \frac{sin ( x ) + cot ( x ) ( 1 + cos ( x ) ) - 2 ( 1 + cos ( x ) )}{1 + cos ( x )}

\frac{sin ( x ) + cot ( x ) ( 1 + cos ( x ) ) - 2 ( 1 + cos ( x ) )}{1 + cos ( x )} = 0

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

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

Express with sin, cos

\frac{sin ^{2} ( x ) + ( 1 + cos ( x ) ) cos ( x ) - ( 1 + cos ( x ) ) \cdot 2 sin ( x )}{sin ( x )} = 0

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

sin^{2} (x) + (1 + cos (x)) cos (x) - (1 + cos (x)) \cdot 2 sin (x) = 0

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

1 - 2 sin (x) = 0 : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

cos (x) + 1 = 0 : x = π + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = π + 2 πn

Since the equation is undefined for:

π + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

4 cos (2 θ) + 16 cos (θ) + 4 = 9 cos (θ)

sin (θ) = \frac{7 π}{3}

tan (x) \cdot cos^{2} (x) - tan (x) = 0

sin (x + π) - sin (x) + 2 = 0

3 tan (x) = - 3

What is the general solution for (sin(x))/(1+cos(x))+cot(x)=2 ?
The general solution for (sin(x))/(1+cos(x))+cot(x)=2 is x= pi/6+2pin,x=(5pi)/6+2pin