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cot^2(a)=cos^2(a)(cot(a)cos(a))^2

Solución

cot^{2} (a) = cos^{2} (a) (cot (a) cos (a))^{2}

Solución

a = \frac{π}{2} + πn

Grados

a = 9 0^{\circ} + 18 0^{\circ} n

Pasos de solución

cot^{2} (a) = cos^{2} (a) (cot (a) cos (a))^{2}

Restar

cos^{2} (a) (cot (a) cos (a))^{2}

de ambos lados

cot^{2} (a) - cos^{4} (a) cot^{2} (a) = 0

Factorizar

cot^{2} (a) - cos^{4} (a) cot^{2} (a) : - cot^{2} (a) (cos^{2} (a) + 1) (cos (a) + 1) (cos (a) - 1)

cot^{2} (a) - cos^{4} (a) cot^{2} (a)

Factorizar el termino común

- cot^{2} (a)

= - cot^{2} (a) (- 1 + cos^{4} (a))

Factorizar

cos^{4} (a) - 1 : (cos^{2} (a) + 1) (cos (a) + 1) (cos (a) - 1)

cos^{4} (a) - 1

Reescribir

cos^{4} (a) - 1

como

(cos^{2} (a))^{2} - 1^{2}

cos^{4} (a) - 1

Reescribir

1

como

1^{2}

= cos^{4} (a) - 1^{2}

Aplicar las leyes de los exponentes:

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

cos^{4} (a) = (cos^{2} (a))^{2}

= (cos^{2} (a))^{2} - 1^{2}

= (cos^{2} (a))^{2} - 1^{2}

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

(cos^{2} (a))^{2} - 1^{2} = (cos^{2} (a) + 1) (cos^{2} (a) - 1)

= (cos^{2} (a) + 1) (cos^{2} (a) - 1)

Factorizar

cos^{2} (a) - 1 : (cos (a) + 1) (cos (a) - 1)

cos^{2} (a) - 1

Reescribir

1

como

1^{2}

= cos^{2} (a) - 1^{2}

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (a) - 1^{2} = (cos (a) + 1) (cos (a) - 1)

= (cos (a) + 1) (cos (a) - 1)

= (cos^{2} (a) + 1) (cos (a) + 1) (cos (a) - 1)

= - cot^{2} (a) (cos^{2} (a) + 1) (cos (a) + 1) (cos (a) - 1)

- cot^{2} (a) (cos^{2} (a) + 1) (cos (a) + 1) (cos (a) - 1) = 0

Resolver cada parte por separado

cot^{2} (a) = 0 or cos^{2} (a) + 1 = 0 or cos (a) + 1 = 0 or cos (a) - 1 = 0

cot^{2} (a) = 0 : a = \frac{π}{2} + πn

cot^{2} (a) = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 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

cos^{2} (a) + 1 = 0 :

Sin solución

cos^{2} (a) + 1 = 0

Usando el método de sustitución

cos^{2} (a) + 1 = 0

Sea:

cos (a) = u

u^{2} + 1 = 0

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

u^{2} + 1 = 0

Mover

1

al lado derecho

u^{2} + 1 = 0

Restar

1

de ambos lados

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

Sustituir en la ecuación

u = cos (a)

cos (a) = i, cos (a) = - i

cos (a) = i, cos (a) = - i

cos (a) = i :

Sin solución

cos (a) = i

Sin soluci \overset{o}{ˊ} n

cos (a) = - i :

Sin solución

cos (a) = - i

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

Sin soluci \overset{o}{ˊ} n

cos (a) + 1 = 0 : a = π + 2 πn

cos (a) + 1 = 0

Mover

1

al lado derecho

cos (a) + 1 = 0

Restar

1

de ambos lados

cos (a) + 1 - 1 = 0 - 1

Simplificar

cos (a) = - 1

cos (a) = - 1

Soluciones generales para

cos (a) = - 1

cos (x)

tabla de valores periódicos con

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = π + 2 πn

a = π + 2 πn

cos (a) - 1 = 0 : a = 2 πn

cos (a) - 1 = 0

Mover

1

al lado derecho

cos (a) - 1 = 0

Sumar

1

a ambos lados

cos (a) - 1 + 1 = 0 + 1

Simplificar

cos (a) = 1

cos (a) = 1

Soluciones generales para

cos (a) = 1

cos (x)

tabla de valores periódicos con

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

a = 0 + 2 πn

a = 0 + 2 πn

Resolver

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn

Combinar toda las soluciones

a = \frac{π}{2} + πn, a = π + 2 πn, a = 2 πn

Siendo que la ecuación esta indefinida para:

π + 2 πn, 2 πn

a = \frac{π}{2} + πn

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