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espandere cot(1/2 (x+y))cot(1/2 (x-y))

Soluzione

espandere

cot (\frac{1}{2} (x + y)) cot (\frac{1}{2} (x - y))

cot (\frac{x + y}{2}) cot (\frac{x - y}{2})

Fasi della soluzione

cot (\frac{1}{2} (x + y)) cot (\frac{1}{2} (x - y))

Espandi

\frac{1}{2} (x + y) : \frac{1}{2} x + \frac{1}{2} y

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

Espandi

\frac{1}{2} (x - y) : \frac{1}{2} x - \frac{1}{2} y

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

Riscrivere utilizzando identità trigonometriche

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

Riscrivere utilizzando identità trigonometriche

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

Moltiplica le frazioni:

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

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

Usa la formula della somma degli angoli:

sin (s) cos (t) + cos (s) sin (t) = sin (s + t)

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

Semplifica

sin (\frac{x}{2} + \frac{y}{2}) (sin (\frac{x}{2}) cos (\frac{y}{2}) - cos (\frac{x}{2}) sin (\frac{y}{2})) : sin (\frac{x}{2} + \frac{y}{2}) sin (\frac{x}{2} - \frac{y}{2})

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

Semplifica

(cos (\frac{x}{2}) cos (\frac{y}{2}) - sin (\frac{x}{2}) sin (\frac{y}{2})) (cos (\frac{x}{2}) cos (\frac{y}{2}) + sin (\frac{x}{2}) sin (\frac{y}{2})) : cos (\frac{x}{2} + \frac{y}{2}) (cos (\frac{x}{2}) cos (\frac{y}{2}) + sin (\frac{x}{2}) sin (\frac{y}{2}))

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

Semplifica

cos (\frac{x}{2}) cos (\frac{y}{2}) + sin (\frac{x}{2}) sin (\frac{y}{2}) : cos (\frac{x}{2} - \frac{y}{2})

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

Combinare le frazioni

\frac{x}{2} - \frac{y}{2} : \frac{x - y}{2}

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

Combinare le frazioni

\frac{x}{2} + \frac{y}{2} : \frac{x + y}{2}

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

Combinare le frazioni

\frac{x}{2} - \frac{y}{2} : \frac{x - y}{2}

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

Combinare le frazioni

\frac{x}{2} + \frac{y}{2} : \frac{x + y}{2}

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

Usare l'identità trigonometrica di base:

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

= cot (\frac{x + y}{2}) cot (\frac{x - y}{2})

e x p an d (1 + e^{2 y}) (e^{y} - \frac{e ^{2 y}}{2})

e x p an d 8 \cdot \frac{d}{d t} ((- 9 t - 6) e^{- 3 t} + (4 t + 12) e^{- 5 t})

e x p an d \frac{5 ( x - 1 )}{6}

\frac{d}{d x} ((z) (z^{3} - 3 x^{2} y + 6 x y z))

e x p an d (M - N)^{2} \cdot e^{- 2 k x} + 4 MN \cdot e^{- k x}