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1/((1+tan(a)))=cot(a)

Solución

\frac{1}{( 1 + tan ( a ) )} = cot (a)

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

Pasos de solución

\frac{1}{( 1 + tan ( a ) )} = cot (a)

Restar

cot (a)

de ambos lados

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

Simplificar

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

\frac{1}{1 + tan ( a )} - cot (a)

Convertir a fracción:

cot (a) = \frac{cot ( a ) ( 1 + tan ( a ) )}{1 + tan ( a )}

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

1 - cot (a) (1 + tan (a)) = 0

Re-escribir usando identidades trigonométricas

1 - (1 + tan (a)) cot (a)

Utilizar la identidad trigonométrica básica:

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

= 1 - (1 + \frac{1}{cot ( a )}) cot (a)

Simplificar

1 - (1 + \frac{1}{cot ( a )}) cot (a) : - cot (a)

1 - (1 + \frac{1}{cot ( a )}) cot (a)

= 1 - cot (a) (1 + \frac{1}{cot ( a )})

Expandir

- cot (a) (1 + \frac{1}{cot ( a )}) : - cot (a) - 1

- cot (a) (1 + \frac{1}{cot ( a )})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = - cot (a), b = 1, c = \frac{1}{cot ( a )}

= - cot (a) \cdot 1 + (- cot (a)) \frac{1}{cot ( a )}

Aplicar las reglas de los signos

+ (- a) = - a

= - 1 \cdot cot (a) - \frac{1}{cot ( a )} cot (a)

Simplificar

- 1 \cdot cot (a) - \frac{1}{cot ( a )} cot (a) : - cot (a) - 1

- 1 \cdot cot (a) - \frac{1}{cot ( a )} cot (a)

1 \cdot cot (a) = cot (a)

1 \cdot cot (a)

Multiplicar:

1 \cdot cot (a) = cot (a)

= cot (a)

\frac{1}{cot ( a )} cot (a) = 1

\frac{1}{cot ( a )} cot (a)

Multiplicar fracciones:

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

= \frac{1 \cdot cot ( a )}{cot ( a )}

Eliminar los terminos comunes:

cot (a)

= 1

= - cot (a) - 1

= - cot (a) - 1

= 1 - cot (a) - 1

Simplificar

1 - cot (a) - 1 : - cot (a)

1 - cot (a) - 1

Agrupar términos semejantes

= - cot (a) + 1 - 1

1 - 1 = 0

= - cot (a)

= - cot (a)

= - cot (a)

- cot (a) = 0

Dividir ambos lados entre

- 1

- cot (a) = 0

Dividir ambos lados entre

- 1

\frac{- cot ( a )}{- 1} = \frac{0}{- 1}

Simplificar

cot (a) = 0

cot (a) = 0

Soluciones generales para

cot (a) = 0

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

a = \frac{π}{2} + πn

a = \frac{π}{2} + πn

Siendo que la ecuación esta indefinida para:

\frac{π}{2} + πn

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

tan (2 x) - 2 = 3 tan (x)

sin (x - \frac{π}{6}) = 0

2 = \frac{1}{cos ^{2} ( x )}

arccot (x - 2) = arccot (x - 1) + arccot (x)

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