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verificar csc(x)tan^2(x)=(sin(x))/(cos^2(x))

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

csc (x) tan^{2} (x)

Expresar con seno, coseno

csc (x) tan^{2} (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Multiplicar fracciones:

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

= \frac{1 \cdot ( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{sin ( x )}

Multiplicar:

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

= \frac{( \frac{s i n ( x )}{c o s ( x )} ) ^{2}}{sin ( x )}

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

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

Aplicar las leyes de los exponentes:

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

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

= \frac{\frac{s i n ^{2} ( x )}{c o s ^{2} ( x )}}{sin ( x )}

Aplicar las propiedades de las fracciones:

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

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

Eliminar los terminos comunes:

sin (x)

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

cos^{4} (x) - sin^{4} (x) = 2 sin^{2} (x) - 1

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

(csc (x) - cot (x)) (sec (x) 1) = tan (x)

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