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verificar (sec(u)-tan(u))(csc(u)+1)=cot(u)

Solución

verificar

(sec (u) - tan (u)) (csc (u) + 1) = cot (u)

Verdadero

Pasos de solución

(sec (u) - tan (u)) (csc (u) + 1) = cot (u)

Manipular el lado derecho

(sec (u) - tan (u)) (csc (u) + 1)

Expresar con seno, coseno

(1 + csc (u)) (sec (u) - tan (u))

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Simplificar

1 + \frac{1}{sin ( u )}

en una fracción:

\frac{sin ( u ) + 1}{sin ( u )}

1 + \frac{1}{sin ( u )}

Convertir a fracción:

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

= \frac{1 \cdot sin ( u )}{sin ( u )} + \frac{1}{sin ( u )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{1 \cdot sin ( u ) + 1}{sin ( u )}

Multiplicar:

1 \cdot sin (u) = sin (u)

= \frac{sin ( u ) + 1}{sin ( u )}

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

Combinar las fracciones usando el mínimo común denominador:

\frac{1 - sin ( u )}{cos ( u )}

Aplicar la regla

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

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

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

Quitar los parentesis:

(a) = a

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

Multiplicar fracciones:

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

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

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

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

Re-escribir usando identidades trigonométricas

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

Expandir

(1 + sin (u)) (1 - sin (u)) : 1 - sin^{2} (u)

(1 + sin (u)) (1 - sin (u))

Aplicar la siguiente regla para binomios al cuadrado:

(a + b) (a - b) = a^{2} - b^{2}

a = 1, b = sin (u)

= 1^{2} - sin^{2} (u)

Aplicar la regla

1^{a} = 1

1^{2} = 1

= 1 - sin^{2} (u)

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

Utilizar la identidad pitagórica:

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

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

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

Eliminar los terminos comunes:

cos (u)

= \frac{cos ( u )}{sin ( u )}

= \frac{cos ( u )}{sin ( u )}

Utilizar la identidad trigonométrica básica:

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

= cot (u)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

cot (2 θ) = \frac{csc ^{2} ( θ ) - 2}{2 cot ( θ )}

tan (θ + \frac{π}{4}) = \frac{1 + tan ( θ )}{1 - tan ( θ )}

tan (θ) = \frac{1}{cot ( θ )}

cos (x) tan (x) sec (- x) = sin (x) - 1

sin (x) = cos (x - \frac{π}{2})