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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado derecho

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

Expresar con seno, coseno

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

Utilizar la identidad trigonométrica básica:

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

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

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

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

Aplicar la regla

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

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

Sumar elementos similares:

- cos (x) + cos (x) = 0

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

Aplicar la regla

\frac{a}{a} = 1

= 1

= 1

= 1

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

cot (x) sec (x) csc^{2} (x) - cot^{3} (x) sec (x) = csc (x)

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

cos (x) \cdot cot (x) + sin (x) = csc (x)

csc (t) - sin (t) = cot (t) cos (t)

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