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## Homework Statement

Show that the reciprocal cubic of cubic lattice is also cubic.

## Homework Equations

[tex]cos\alpha*=\frac{cos\beta cos\gamma-cos\alpha}{sin\beta sin\gamma}[/tex]

[tex]cos\beta*=\frac{cos\alpha cos\gamma-cos\beta}{sin\alpha sin\gamma}[/tex]

[tex]cos\gamma*=\frac{cos\alpha cos\beta-cos\gamma}{sin\alpha sin\beta}[/tex]

[tex]\vec{a*}=\frac{\vec{b}\times\vec{c}}{V}[/tex]

[tex]\vec{b*}=\frac{\vec{c}\times\vec{a}}{V}[/tex]

[tex]\vec{c*}=\frac{\vec{a}\times\vec{b}}{V}[/tex]

## The Attempt at a Solution

If I use this formula I will show that [tex]\alpha*=\beta*=\gamma*=90^{\circ}[/tex]

and [tex]a*=b*=c*=\frac{1}{a}[/tex]

and so reciprocal lattice of cubic lattice is cubic. Q.E.D.

But I don't know from where I get this angle relations

[tex]cos\alpha*=\frac{cos\beta cos\gamma-cos\alpha}{sin\beta sin\gamma}[/tex]

[tex]cos\beta*=\frac{cos\alpha cos\gamma-cos\beta}{sin\alpha sin\gamma}[/tex]

[tex]cos\gamma*=\frac{cos\alpha cos\beta-cos\gamma}{sin\alpha sin\beta}[/tex]