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verificar (cot(x)+csc(x)cos(x))/(cot(x))=2

Solución

verificar

\frac{cot ( x ) + csc ( x ) cos ( x )}{cot ( x )} = 2

Verdadero

Pasos de solución

\frac{cot ( x ) + csc ( x ) cos ( x )}{cot ( x )} = 2

Manipular el lado derecho

\frac{cot ( x ) + csc ( x ) cos ( x )}{cot ( x )}

Expresar con seno, coseno

\frac{cot ( x ) + cos ( x ) csc ( x )}{cot ( x )}

Utilizar la identidad trigonométrica básica:

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

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

Utilizar la identidad trigonométrica básica:

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

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

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

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

Aplicar las propiedades de las fracciones:

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

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

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

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

Multiplicar fracciones:

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

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

Multiplicar:

1 \cdot cos (x) = cos (x)

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

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

Combinar las fracciones usando el mínimo común denominador:

\frac{2 cos ( 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 )}

Sumar elementos similares:

cos (x) + cos (x) = 2 cos (x)

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

= \frac{( \frac{2 c o s ( x )}{s i n ( x )} ) sin ( x )}{cos ( x )}

Quitar los parentesis:

(a) = a

= \frac{\frac{2 c o s ( x )}{s i n ( x )} sin ( x )}{cos ( x )}

Multiplicar

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

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

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

sin (x)

= 2 cos (x)

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

Eliminar los terminos comunes:

cos (x)

= 2

= 2

= 2

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

sec (θ) - cos (θ) = sin (θ) \cdot tan (θ)

(2 sin (x) + cos (x))^{2} + (2 cos (x) - sin (x))^{2} = 5

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

tan (x) cos (x) csc (x) = 1

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