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verificar csc(2x)-cot(2x)=tan(x)

Solución

verificar

csc (2 x) - cot (2 x) = tan (x)

Verdadero

Pasos de solución

csc (2 x) - cot (2 x) = tan (x)

Manipular el lado derecho

csc (2 x) - cot (2 x)

Expresar con seno, coseno

- cot (2 x) + csc (2 x)

Utilizar la identidad trigonométrica básica:

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

= - \frac{cos ( 2 x )}{sin ( 2 x )} + csc (2 x)

Utilizar la identidad trigonométrica básica:

csc (x) = \frac{1}{sin ( x )}

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

Simplificar

- \frac{cos ( 2 x )}{sin ( 2 x )} + \frac{1}{sin ( 2 x )} : \frac{- cos ( 2 x ) + 1}{sin ( 2 x )}

- \frac{cos ( 2 x )}{sin ( 2 x )} + \frac{1}{sin ( 2 x )}

Aplicar la regla

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

Expandir

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

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

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

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

Poner los parentesis

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

Aplicar las reglas de los signos

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

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

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

1 - 1 = 0

= 2 sin^{2} (x)

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

Dividir:

\frac{2}{2} = 1

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

Eliminar los terminos comunes:

sin (x)

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

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

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

Utilizar la identidad trigonométrica básica:

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

= tan (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero