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verificar 1/(sec(x)(1-sin(x)))=sec(x)+tan(x)

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado izquierdo

sec (x) + tan (x)

Expresar con seno, coseno

sec (x) + tan (x)

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

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

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

Aplicar la regla

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

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

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

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

Re-escribir usando identidades trigonométricas

Multiplicar

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

por

Eq u a t i o n 1

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

Utilizar la identidad trigonométrica básica:

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

\frac{\frac{1}{s e c ( x )}}{1 - sin ( x )}

Simplificar

\frac{\frac{1}{s e c ( x )}}{1 - sin ( x )}

Aplicar las propiedades de las fracciones:

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

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

\frac{1}{sec ( x ) ( 1 - sin ( x ) )}

\frac{1}{sec ( x ) ( 1 - sin ( x ) )}

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

sin^{2} (y) cos^{2} (y) = \frac{1 - cos ( 4 x )}{8}

cos (θ) tan (θ) sin (θ) = sin^{2} (θ)

cos (\frac{3 x}{5}) = - sin (\frac{3 x}{5} - \frac{π}{2})

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