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

Solución

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

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

Pasos de solución

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

Restar

2

de ambos lados

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

Simplificar

\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

Expresar con seno, coseno

\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

Re-escribir usando identidades trigonométricas

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

Factorizar

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

Resolver cada parte por separado

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

Combinar toda las soluciones

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

Siendo que la ecuación esta indefinida para:

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