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verificar sin(x)csc^2(x)= 1/(sin(x))

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

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

Expresar con seno, coseno

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar la regla

1^{a} = 1

1^{2} = 1

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

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

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot sin (x) = sin (x)

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

Eliminar los terminos comunes:

sin (x)

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

\frac{cos ( 2 θ )}{cos ^{2} ( θ )} = 1 - tan^{2} (θ)

(tan (θ + 1)) = sec (θ)

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

tan (x) + sec (x) = 1

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