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verificar (sin(x)csc(x))/(cos(x))=sec(x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

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

Expresar con seno, coseno

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

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{1}{s i n ( x )} sin ( x )}{cos ( x )}

Simplificar

\frac{\frac{1}{s i n ( x )} sin ( x )}{cos ( x )} : \frac{1}{cos ( x )}

\frac{\frac{1}{s i n ( x )} sin ( x )}{cos ( x )}

Multiplicar

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

sin (x)

= 1

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

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

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

Re-escribir usando identidades trigonométricas

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

Aplicar las propiedades de las fracciones:

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

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

Aplicar la regla

\frac{a}{1} = a

= sec (x)

sec (x)

sec (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

tan (x) + sec (x) = 1

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

sin^{2} (x) + cos^{2} (x) - 1 = 0

sec (x) sin (\frac{π}{2} - x) = 1

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