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verificar cos(x)-csc(x)cot(x)=-cos(x)cot^2(x)

Solución

verificar

cos (x) - csc (x) cot (x) = - cos (x) cot^{2} (x)

Verdadero

Pasos de solución

cos (x) - csc (x) cot (x) = - cos (x) cot^{2} (x)

Manipular el lado derecho

cos (x) - csc (x) cot (x)

Expresar con seno, coseno

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

Factorizar

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

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

Re-escribir usando identidades trigonométricas

= - \frac{cos ^{3} ( x )}{sin ^{2} ( x )}

Re-escribir usando identidades trigonométricas

- cot^{2} (x) cos (x)

= - cos (x) cot^{2} (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

\frac{1 + tan ( x )}{sin ( x )} - sec (x) = csc (x)

\frac{csc ( β )}{cot ( β )} = sec (β)

\frac{cot ( θ ) - csc ( θ )}{cot ( θ ) + csc ( θ )} = \frac{1 - 2 cos ( θ ) + cos ^{2} ( θ )}{- sin ^{2} ( θ )}

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