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verificar (1-cos(x))/(sin(x))=csc(x)-cot(x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado izquierdo

csc (x) - cot (x)

Expresar con seno, coseno

- cot (x) + csc (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Aplicar la regla

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

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

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

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

csc (2 x) = \frac{1}{2} sec (x) csc (x)

\frac{1}{1 - tan ^{2} ( x )} + \frac{1}{1 - cot ^{2} ( x )} = 1

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

csc (x) - cot (x) = \frac{1}{csc ( x ) + cot ( x )}