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verificar sin^2(θ/2)=(csc(θ)-cot(θ))/(2csc(θ))

Solución

verificar

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

Solución

Verdadero

Pasos de solución

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

Sea:

u = \frac{θ}{2}

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

Verificar

sin^{2} (u) = \frac{csc ( 2 u ) - cot ( 2 u )}{2 csc ( 2 u )} :

Verdadero

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

Manipular el lado izquierdo

\frac{csc ( 2 u ) - cot ( 2 u )}{2 csc ( 2 u )}

Expresar con seno, coseno

\frac{- cot ( 2 u ) + csc ( 2 u )}{2 csc ( 2 u )}

Utilizar la identidad trigonométrica básica:

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

= \frac{- \frac{c o s ( 2 u )}{s i n ( 2 u )} + csc ( 2 u )}{2 csc ( 2 u )}

Utilizar la identidad trigonométrica básica:

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

= \frac{- \frac{c o s ( 2 u )}{s i n ( 2 u )} + \frac{1}{s i n ( 2 u )}}{2 \cdot \frac{1}{s i n ( 2 u )}}

Simplificar

\frac{- \frac{c o s ( 2 u )}{s i n ( 2 u )} + \frac{1}{s i n ( 2 u )}}{2 \cdot \frac{1}{s i n ( 2 u )}} : \frac{- cos ( 2 u ) + 1}{2}

\frac{- \frac{c o s ( 2 u )}{s i n ( 2 u )} + \frac{1}{s i n ( 2 u )}}{2 \cdot \frac{1}{s i n ( 2 u )}}

Combinar las fracciones usando el mínimo común denominador:

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

Aplicar la regla

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

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

= \frac{\frac{- c o s ( 2 u ) + 1}{s i n ( 2 u )}}{2 \cdot \frac{1}{s i n ( 2 u )}}

Aplicar las propiedades de las fracciones:

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

= \frac{- cos ( 2 u ) + 1}{sin ( 2 u ) \cdot 2 \cdot \frac{1}{s i n ( 2 u )}}

Multiplicar

sin (2 u) \cdot 2 \cdot \frac{1}{sin ( 2 u )} : 2

sin (2 u) \cdot 2 \cdot \frac{1}{sin ( 2 u )}

Multiplicar fracciones:

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

= \frac{1 \cdot sin ( 2 u ) \cdot 2}{sin ( 2 u )}

Eliminar los terminos comunes:

sin (2 u)

= 1 \cdot 2

Multiplicar los numeros:

1 \cdot 2 = 2

= 2

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

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

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

Re-escribir usando identidades trigonométricas

\frac{1 - cos ( 2 u )}{2}

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

Expandir

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

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

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

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

Poner los parentesis

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

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

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

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

1 - 1 = 0

= 2 sin^{2} (u)

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

Dividir:

\frac{2}{2} = 1

= sin^{2} (u)

= sin^{2} (u)

= sin^{2} (u)

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Por lo tanto

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

Se demostró que ambos lados pueden tomar la misma forma

\Rightarrow Verdadero

Ejemplos populares

verificar cos(2x)=(cot^2(x)-1)/(cot^2(x)+1)verificar

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

verificar sin^2(x)-2cos^2(x)=3sin^2(x)-2 verificar

sin^{2} (x) - 2 cos^{2} (x) = 3 sin^{2} (x) - 2

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

cos (x) + cos (x) cot^{2} (x) = cot (x) csc (x)

verificar tan(a)=tan(a)csc^2(a)+cot(-a)verificar

tan (a) = tan (a) csc^{2} (a) + cot (- a)

verificar 7csc^2(x)-5cot^2(x)=2csc^2(x)+5 verificar

7 csc^{2} (x) - 5 cot^{2} (x) = 2 csc^{2} (x) + 5