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verificar (sec(x))/(csc(x))+(sin(x))/(cos(x))=2tan(x)

Solución

verificar

\frac{sec ( x )}{csc ( x )} + \frac{sin ( x )}{cos ( x )} = 2 tan (x)

Verdadero

Pasos de solución

\frac{sec ( x )}{csc ( x )} + \frac{sin ( x )}{cos ( x )} = 2 tan (x)

Manipular el lado derecho

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

Expresar con seno, coseno

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

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{1}{c o s ( x )}}{csc ( x )} + \frac{sin ( x )}{cos ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{1}{c o s ( x )}}{\frac{1}{s i n ( x )}} + \frac{sin ( x )}{cos ( x )}

Simplificar

\frac{\frac{1}{c o s ( x )}}{\frac{1}{s i n ( x )}} + \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

\frac{\frac{1}{c o s ( x )}}{\frac{1}{s i n ( x )}} + \frac{sin ( x )}{cos ( x )}

\frac{\frac{1}{c o s ( x )}}{\frac{1}{s i n ( x )}} = \frac{sin ( x )}{cos ( x )}

\frac{\frac{1}{c o s ( x )}}{\frac{1}{s i n ( x )}}

Dividir fracciones:

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

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

Simplificar

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

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

Aplicar la regla

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

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

Sumar elementos similares:

sin (x) + sin (x) = 2 sin (x)

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

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

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

Re-escribir usando identidades trigonométricas

= 2 \cdot \frac{sin ( x )}{cos ( x )}

Utilizar la identidad trigonométrica básica:

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

2 tan (x)

2 tan (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

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

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

cos (2 a) = cos^{2} (a) - sin^{2} (a)

tan (θ) + cot (θ) = sec (θ) \cdot csc (θ)

\frac{cos ( x )}{csc ( x ) + 1} + \frac{cos ( x )}{csc ( x ) - 1} = 2 tan (x)