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verificar 2csc(2x)=sec(x)csc(x)

Solución

verificar

2 csc (2 x) = sec (x) csc (x)

Verdadero

Pasos de solución

2 csc (2 x) = sec (x) csc (x)

Manipular el lado derecho

2 csc (2 x)

Expresar con seno, coseno

2 csc (2 x)

Utilizar la identidad trigonométrica básica:

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

= 2 \cdot \frac{1}{sin ( 2 x )}

Simplificar

2 \cdot \frac{1}{sin ( 2 x )} : \frac{2}{sin ( 2 x )}

2 \cdot \frac{1}{sin ( 2 x )}

Multiplicar fracciones:

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

= \frac{1 \cdot 2}{sin ( 2 x )}

Multiplicar los numeros:

1 \cdot 2 = 2

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

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

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

Re-escribir usando identidades trigonométricas

\frac{2}{sin ( 2 x )}

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Dividir:

\frac{2}{2} = 1

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

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

Re-escribir usando identidades trigonométricas

Utilizar la identidad trigonométrica básica:

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

\frac{1}{\frac{1}{c s c ( x )} cos ( x )}

Utilizar la identidad trigonométrica básica:

cos (x) = \frac{1}{sec ( x )}

\frac{1}{\frac{1}{c s c ( x )} \cdot \frac{1}{s e c ( x )}}

Simplificar

\frac{1}{\frac{1}{c s c ( x )} \cdot \frac{1}{s e c ( x )}}

Multiplicar

\frac{1}{csc ( x )} \cdot \frac{1}{sec ( x )} : \frac{1}{csc ( x ) sec ( x )}

\frac{1}{csc ( x )} \cdot \frac{1}{sec ( x )}

Multiplicar fracciones:

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

= \frac{1 \cdot 1}{csc ( x ) sec ( x )}

Multiplicar los numeros:

1 \cdot 1 = 1

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

= \frac{1}{\frac{1}{c s c ( x ) s e c ( x )}}

Aplicar las propiedades de las fracciones:

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

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

Aplicar la regla

\frac{a}{1} = a

= csc (x) sec (x)

csc (x) sec (x)

csc (x) sec (x)

= sec (x) csc (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

(tan (θ + 1)) = sec (θ)

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

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

tan (x) + sec (x) = 1

- sinh^{2} (x) + cosh^{2} (x) = 1