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(1-csc(θ))/(cot(θ))=tan(θ)-csc(θ)

Solución

\frac{1 - csc ( θ )}{cot ( θ )} = tan (θ) - csc (θ)

θ = \frac{π}{4} + πn

Grados

θ = 4 5^{\circ} + 18 0^{\circ} n

Pasos de solución

\frac{1 - csc ( θ )}{cot ( θ )} = tan (θ) - csc (θ)

Restar

tan (θ) - csc (θ)

de ambos lados

\frac{1 - csc ( θ )}{cot ( θ )} - tan (θ) + csc (θ) = 0

Simplificar

\frac{1 - csc ( θ )}{cot ( θ )} - tan (θ) + csc (θ) : \frac{1 - csc ( θ ) - tan ( θ ) cot ( θ ) + csc ( θ ) cot ( θ )}{cot ( θ )}

\frac{1 - csc ( θ )}{cot ( θ )} - tan (θ) + csc (θ)

Convertir a fracción:

tan (θ) = \frac{tan ( θ ) cot ( θ )}{cot ( θ )}, csc (θ) = \frac{csc ( θ ) cot ( θ )}{cot ( θ )}

= \frac{1 - csc ( θ )}{cot ( θ )} - \frac{tan ( θ ) cot ( θ )}{cot ( θ )} + \frac{csc ( θ ) cot ( θ )}{cot ( θ )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{1 - csc ( θ ) - tan ( θ ) cot ( θ ) + csc ( θ ) cot ( θ )}{cot ( θ )}

\frac{1 - csc ( θ ) - tan ( θ ) cot ( θ ) + csc ( θ ) cot ( θ )}{cot ( θ )} = 0

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

1 - csc (θ) - tan (θ) cot (θ) + csc (θ) cot (θ) = 0

Re-escribir usando identidades trigonométricas

1 - csc (θ) + cot (θ) csc (θ) - cot (θ) tan (θ)

Utilizar la identidad trigonométrica básica:

tan (x) = \frac{1}{cot ( x )}

= 1 - csc (θ) + cot (θ) csc (θ) - cot (θ) \frac{1}{cot ( θ )}

Simplificar

1 - csc (θ) + cot (θ) csc (θ) - cot (θ) \frac{1}{cot ( θ )} : cot (θ) csc (θ) - csc (θ)

1 - csc (θ) + cot (θ) csc (θ) - cot (θ) \frac{1}{cot ( θ )}

cot (θ) \frac{1}{cot ( θ )} = 1

cot (θ) \frac{1}{cot ( θ )}

Multiplicar fracciones:

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

= \frac{1 \cdot cot ( θ )}{cot ( θ )}

Eliminar los terminos comunes:

cot (θ)

= 1

= 1 - csc (θ) + cot (θ) csc (θ) - 1

Agrupar términos semejantes

= - csc (θ) + cot (θ) csc (θ) + 1 - 1

1 - 1 = 0

= cot (θ) csc (θ) - csc (θ)

= cot (θ) csc (θ) - csc (θ)

- csc (θ) + cot (θ) csc (θ) = 0

Factorizar

- csc (θ) + cot (θ) csc (θ) : csc (θ) (cot (θ) - 1)

- csc (θ) + cot (θ) csc (θ)

Factorizar el termino común

csc (θ)

= csc (θ) (- 1 + cot (θ))

csc (θ) (cot (θ) - 1) = 0

Resolver cada parte por separado

csc (θ) = 0 or cot (θ) - 1 = 0

csc (θ) = 0 :

Sin solución

csc (θ) = 0

csc (x) \leq - 1 or csc (x) \geq 1

Sin soluci \overset{o}{ˊ} n

cot (θ) - 1 = 0 : θ = \frac{π}{4} + πn

cot (θ) - 1 = 0

Mover

1

al lado derecho

cot (θ) - 1 = 0

Sumar

1

a ambos lados

cot (θ) - 1 + 1 = 0 + 1

Simplificar

cot (θ) = 1

cot (θ) = 1

Soluciones generales para

cot (θ) = 1

cot (x)

tabla de valores periódicos con

πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

θ = \frac{π}{4} + πn

θ = \frac{π}{4} + πn

Combinar toda las soluciones

θ = \frac{π}{4} + πn

2 sin (a) - 1 = 0

cos (3 x) = cos (2 x) cos (x)

3 cos^{2} (x) + sin^{2} (x) + 5 sin (x) = 0

3 cos (θ) = 2 - sin (θ)

2 cos^{2} (θ) sin (θ) + cos^{2} (θ) = 0