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

Solución

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

Sin soluci \overset{o}{ˊ} n para x \in R

Pasos de solución

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

Restar

1 + cos (x)

de ambos lados

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

Simplificar

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

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

Convertir a fracción:

1 = \frac{1 sin ( x )}{sin ( x )}, cos (x) = \frac{cos ( x ) sin ( x )}{sin ( x )}

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

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

Sumar elementos similares:

sin (x) - 1 \cdot sin (x) = 0

= \frac{1 - cos ( x ) sin ( x )}{sin ( x )}

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

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

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

Re-escribir usando identidades trigonométricas

1 - cos (x) sin (x)

Utilizar la identidad trigonométrica del ángulo doble:

2 sin (x) cos (x) = sin (2 x)

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

= 1 - \frac{sin ( 2 x )}{2}

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

Mover

1

al lado derecho

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

Restar

1

de ambos lados

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

Simplificar

- \frac{sin ( 2 x )}{2} = - 1

- \frac{sin ( 2 x )}{2} = - 1

Multiplicar ambos lados por

2

- \frac{sin ( 2 x )}{2} = - 1

Multiplicar ambos lados por

2

2 (- \frac{sin ( 2 x )}{2}) = 2 (- 1)

Simplificar

- sin (2 x) = - 2

- sin (2 x) = - 2

Dividir ambos lados entre

- 1

- sin (2 x) = - 2

Dividir ambos lados entre

- 1

\frac{- sin ( 2 x )}{- 1} = \frac{- 2}{- 1}

Simplificar

sin (2 x) = 2

sin (2 x) = 2

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n para x \in R

cos (3 x) - cos (5 x) = 0

2 = 2 cos (x)

4 cos (x) = 4 cos (2 x)

cos (θ) = 513, 3 π^{2} < θ < 2 π

cos (x) + ∣ cos (x) ∣ = 1