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

Solución

verificar

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

Verdadero

Pasos de solución

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

Manipular el lado izquierdo

\frac{csc ( x ) cot ( x )}{csc ^{2} ( x )}

Expresar con seno, coseno

\frac{cot ( x ) csc ( x )}{csc ^{2} ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{c o s ( x )}{s i n ( x )} csc ( x )}{csc ^{2} ( x )}

Utilizar la identidad trigonométrica básica:

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

= \frac{\frac{c o s ( x )}{s i n ( x )} \cdot \frac{1}{s i n ( x )}}{( \frac{1}{s i n ( x )} ) ^{2}}

Simplificar

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

\frac{\frac{c o s ( x )}{s i n ( x )} \cdot \frac{1}{s i n ( x )}}{( \frac{1}{s i n ( x )} ) ^{2}}

(\frac{1}{sin ( x )})^{2} = \frac{1}{sin ^{2} ( x )}

(\frac{1}{sin ( x )})^{2}

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{1 ^{2}}{sin ^{2} ( x )}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( x )}

= \frac{\frac{c o s ( x )}{s i n ( x )} \cdot \frac{1}{s i n ( x )}}{\frac{1}{s i n ^{2} ( x )}}

Multiplicar

\frac{cos ( x )}{sin ( x )} \cdot \frac{1}{sin ( x )} : \frac{cos ( x )}{sin ^{2} ( x )}

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

Multiplicar fracciones:

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

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

Multiplicar:

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

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

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

sin (x) sin (x) = sin^{1 + 1} (x)

= sin^{1 + 1} (x)

Sumar:

1 + 1 = 2

= sin^{2} (x)

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

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

Dividir fracciones:

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

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

Simplificar

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

Eliminar los terminos comunes:

sin^{2} (x)

= cos (x)

= cos (x)

= cos (x)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

csc^{2} (\frac{x}{2}) = \frac{2 sec ( x )}{sec ( x ) - 1}

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

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

\frac{sin ^{2} ( x )}{1} \cdot \frac{1}{sin ^{2} ( x )} = 1

\frac{1}{2} cot (x) - \frac{1}{2} tan (x) = cot (2 x)