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Popular Trigonometría >

(1-tan^2(A))/(2tan(A))=cot(A)

Solución

\frac{1 - tan ^{2} ( A )}{2 tan ( A )} = cot (A)

Solución

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

Pasos de solución

\frac{1 - tan ^{2} ( A )}{2 tan ( A )} = cot (A)

Restar

cot (A)

de ambos lados

\frac{1 - tan ^{2} ( A )}{2 tan ( A )} - cot (A) = 0

Simplificar

\frac{1 - tan ^{2} ( A )}{2 tan ( A )} - cot (A) : \frac{1 - tan ^{2} ( A ) - 2 cot ( A ) tan ( A )}{2 tan ( A )}

\frac{1 - tan ^{2} ( A )}{2 tan ( A )} - cot (A)

Convertir a fracción:

cot (A) = \frac{cot ( A ) 2 tan ( A )}{2 tan ( A )}

= \frac{1 - tan ^{2} ( A )}{2 tan ( A )} - \frac{cot ( A ) \cdot 2 tan ( A )}{2 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 - tan ^{2} ( A ) - cot ( A ) \cdot 2 tan ( A )}{2 tan ( A )}

\frac{1 - tan ^{2} ( A ) - 2 cot ( A ) tan ( A )}{2 tan ( A )} = 0

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

1 - tan^{2} (A) - 2 cot (A) tan (A) = 0

Re-escribir usando identidades trigonométricas

1 - tan^{2} (A) - 2 cot (A) tan (A)

Utilizar la identidad trigonométrica básica:

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

= 1 - (\frac{1}{cot ( A )})^{2} - 2 cot (A) \frac{1}{cot ( A )}

Simplificar

1 - (\frac{1}{cot ( A )})^{2} - 2 cot (A) \frac{1}{cot ( A )} : - \frac{1}{cot ^{2} ( A )} - 1

1 - (\frac{1}{cot ( A )})^{2} - 2 cot (A) \frac{1}{cot ( A )}

(\frac{1}{cot ( A )})^{2} = \frac{1}{cot ^{2} ( A )}

(\frac{1}{cot ( A )})^{2}

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{cot ^{2} ( A )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{cot ^{2} ( A )}

2 cot (A) \frac{1}{cot ( A )} = 2

2 cot (A) \frac{1}{cot ( A )}

Multiplicar fracciones:

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

= \frac{1 \cdot 2 cot ( A )}{cot ( A )}

Eliminar los terminos comunes:

cot (A)

= 1 \cdot 2

Multiplicar los numeros:

1 \cdot 2 = 2

= 2

= 1 - \frac{1}{cot ^{2} ( A )} - 2

Agrupar términos semejantes

= - \frac{1}{cot ^{2} ( A )} + 1 - 2

Sumar/restar lo siguiente:

1 - 2 = - 1

= - \frac{1}{cot ^{2} ( A )} - 1

= - \frac{1}{cot ^{2} ( A )} - 1

- 1 - \frac{1}{cot ^{2} ( A )} = 0

Usando el método de sustitución

- 1 - \frac{1}{cot ^{2} ( A )} = 0

Sea:

cot (A) = u

- 1 - \frac{1}{u ^{2}} = 0

- 1 - \frac{1}{u ^{2}} = 0 : u = i, u = - i

- 1 - \frac{1}{u ^{2}} = 0

Multiplicar ambos lados por

u^{2}

- 1 - \frac{1}{u ^{2}} = 0

Multiplicar ambos lados por

u^{2}

- 1 \cdot u^{2} - \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplificar

- 1 \cdot u^{2} - \frac{1}{u ^{2}} u^{2} = 0 \cdot u^{2}

Simplificar

- 1 \cdot u^{2} : - u^{2}

- 1 \cdot u^{2}

Multiplicar:

1 \cdot u^{2} = u^{2}

= - u^{2}

Simplificar

- \frac{1}{u ^{2}} u^{2} : - 1

- \frac{1}{u ^{2}} u^{2}

Multiplicar fracciones:

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

= - \frac{1 \cdot u ^{2}}{u ^{2}}

Eliminar los terminos comunes:

u^{2}

= - 1

Simplificar

0 \cdot u^{2} : 0

0 \cdot u^{2}

Aplicar la regla

0 \cdot a = 0

= 0

- u^{2} - 1 = 0

- u^{2} - 1 = 0

- u^{2} - 1 = 0

Resolver

- u^{2} - 1 = 0 : u = i, u = - i

- u^{2} - 1 = 0

Desplace

1

a la derecha

- u^{2} - 1 = 0

Sumar

1

a ambos lados

- u^{2} - 1 + 1 = 0 + 1

Simplificar

- u^{2} = 1

- u^{2} = 1

Dividir ambos lados entre

- 1

- u^{2} = 1

Dividir ambos lados entre

- 1

\frac{- u ^{2}}{- 1} = \frac{1}{- 1}

Simplificar

u^{2} = - 1

u^{2} = - 1

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = - 1, u = - - 1

Simplificar

- 1 : i

- 1

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= i

Simplificar

- - 1 : - i

- - 1

Aplicar las propiedades de los numeros imaginarios:

- 1 = i

= - i

u = i, u = - i

u = i, u = - i

Sustituir en la ecuación

u = cot (A)

cot (A) = i, cot (A) = - i

cot (A) = i, cot (A) = - i

cot (A) = i :

Sin solución

cot (A) = i

Sin soluci \overset{o}{ˊ} n

cot (A) = - i :

Sin solución

cot (A) = - i

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

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

Gráfica

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