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verificar 1/(sin(x))-1/(tan(x))=tan(1/2 x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Sea:

u = \frac{1}{2} x

\frac{1}{sin ( 2 u )} - \frac{1}{tan ( 2 u )} = tan (u)

Verificar

\frac{1}{sin ( 2 u )} - \frac{1}{tan ( 2 u )} = tan (u) :

Verdadero

\frac{1}{sin ( 2 u )} - \frac{1}{tan ( 2 u )} = tan (u)

Manipular el lado derecho

\frac{1}{sin ( 2 u )} - \frac{1}{tan ( 2 u )}

Expresar con seno, coseno

\frac{1}{sin ( 2 u )} - \frac{1}{tan ( 2 u )}

Utilizar la identidad trigonométrica básica:

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

= \frac{1}{sin ( 2 u )} - \frac{1}{\frac{s i n ( 2 u )}{c o s ( 2 u )}}

Simplificar

\frac{1}{sin ( 2 u )} - \frac{1}{\frac{s i n ( 2 u )}{c o s ( 2 u )}} : \frac{1 - cos ( 2 u )}{sin ( 2 u )}

\frac{1}{sin ( 2 u )} - \frac{1}{\frac{s i n ( 2 u )}{c o s ( 2 u )}}

Aplicar las propiedades de las fracciones:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{1}{sin ( 2 u )} - \frac{cos ( 2 u )}{sin ( 2 u )}

Aplicar la regla

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

= \frac{1 - cos ( 2 u )}{sin ( 2 u )}

= \frac{1 - cos ( 2 u )}{sin ( 2 u )}

= \frac{1 - cos ( 2 u )}{sin ( 2 u )}

Re-escribir usando identidades trigonométricas

\frac{1 - cos ( 2 u )}{sin ( 2 u )}

Utilizar la identidad trigonométrica del ángulo doble:

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

= \frac{1 - cos ( 2 u )}{2 sin ( u ) cos ( u )}

Utilizar la identidad trigonométrica del ángulo doble:

cos (2 x) = 1 - 2 sin^{2} (x)

= \frac{1 - ( 1 - 2 sin ^{2} ( u ) )}{2 sin ( u ) cos ( u )}

Simplificar

\frac{1 - ( 1 - 2 sin ^{2} ( u ) )}{2 sin ( u ) cos ( u )} : \frac{sin ( u )}{cos ( u )}

\frac{1 - ( 1 - 2 sin ^{2} ( u ) )}{2 sin ( u ) cos ( u )}

Expandir

1 - (1 - 2 sin^{2} (u)) : 2 sin^{2} (u)

1 - (1 - 2 sin^{2} (u))

- (1 - 2 sin^{2} (u)) : - 1 + 2 sin^{2} (u)

- (1 - 2 sin^{2} (u))

Poner los parentesis

= - (1) - (- 2 sin^{2} (u))

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

= - 1 + 2 sin^{2} (u)

= 1 - 1 + 2 sin^{2} (u)

1 - 1 = 0

= 2 sin^{2} (u)

= \frac{2 sin ^{2} ( u )}{2 sin ( u ) cos ( u )}

Dividir:

\frac{2}{2} = 1

= \frac{sin ^{2} ( u )}{sin ( u ) cos ( u )}

Eliminar los terminos comunes:

sin (u)

= \frac{sin ( u )}{cos ( u )}

= \frac{sin ( u )}{cos ( u )}

= \frac{sin ( u )}{cos ( u )}

Utilizar la identidad trigonométrica básica:

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

= tan (u)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Por lo tanto

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

tan^{2} (x) = - 1 + sec^{2} (x)

- tan^{2} (x) = 1 - sec^{2} (x)

\frac{cot ^{2} ( x )}{csc ^{2} ( x )} = cos^{2} (x)

(cos (x) - sin (x)) 2 = 1 - 2 sin (x) cos (x)

cos (x) (csc^{2} (x)) = cot (x) csc (x)