Identidades trigonométricas

[latex]\mathrm{M=\frac{sin(40^{\circ})}{sin(80^{\circ})}+\frac{sin(80^{\circ})}{sin(20^{\circ})}-\frac{sin(20^{\circ})}{sin(40^{\circ})}}[/latex]

[latex]\mathrm{M=\frac{sin(40^{\circ})}{2sin(40^{\circ})cos(40^{\circ})}+\frac{2sin(40^{\circ})cos(40^{\circ})}{sin(20^{\circ})}-\frac{sin(20^{\circ})}{2sin(20^{\circ})cos(20^{\circ})}}[/latex]

[latex]\mathrm{M=\frac{1}{2cos(40^{\circ})}+\frac{4sin(20^{\circ})cos(20^{\circ})cos(40^{\circ})}{sin(20^{\circ})}-\frac{1}{2cos(20^{\circ})}}[/latex]

[latex]\mathrm{M=\frac{1}{2cos(40^{\circ})}+2[2cos(20^{\circ})cos(40^{\circ})]-\frac{1}{2cos(20^{\circ})}}[/latex]

[latex]\mathrm{M=\frac{1}{2}\left [ \frac{1}{cos(40^{\circ})}-\frac{1}{cos(20^{\circ})} \right ]+\underset{Prostaf\acute{e}rese}{\underbrace{\mathrm{2[cos(60^{\circ})+cos(20^{\circ})]}}}}[/latex]

[latex]\mathrm{M=\frac{1}{2}\left [ \frac{cos(20^{\circ})-cos(40^{\circ})}{cos(20^{\circ})cos(40^{\circ})} \right ]+2[cos(60^{\circ})+cos(20^{\circ})]}[/latex]

[latex]\mathrm{M=\frac{2sin(30^{\circ})sin(10^{\circ})}{2cos(20^{\circ})cos(40^{\circ})}+2cos(20^{\circ})+1}[/latex]

[latex]\mathrm{M=\frac{sin(10^{\circ})}{2cos(20^{\circ})cos(40^{\circ})}+2cos(20^{\circ})+1}[/latex]

[latex]\mathrm{M=\frac{2sin(20^{\circ})sin(10^{\circ})}{4sin(20^{\circ})cos(20^{\circ})cos(40^{\circ})}+2cos(20^{\circ})+1}[/latex]

[latex]\mathrm{M=\frac{2sin(20^{\circ})sin(10^{\circ})}{4sin(20^{\circ})cos(20^{\circ})cos(40^{\circ})}+2cos(20^{\circ})+1}[/latex]

[latex]\mathrm{M=\frac{2sin(20^{\circ})sin(10^{\circ})}{2sin(40^{\circ})cos(40^{\circ})}+2cos(20^{\circ})+1}[/latex]

[latex]\mathrm{M=\frac{2sin(10^{\circ})sin(20^{\circ})+2sin(80^{\circ})cos(20^{\circ})}{sin(80^{\circ})}+1}[/latex]

[latex]\mathrm{Sendo\ sin(x)=cos(90^{\circ}-x)\ \therefore \ M=\frac{2sin(10^{\circ})sin(20^{\circ})+2cos(10^{\circ})cos(20^{\circ})}{sin(80^{\circ})}+1}[/latex]

[latex]\mathrm{M=\frac{cos(10^{\circ})-cos(30^{\circ})+cos(30^{\circ})+cos(10^{\circ})}{sin(80^{\circ})}+1=\frac{2cos(10^{\circ})}{sin(80^{\circ})}=\frac{2sin(80^{\circ})}{sin(80^{\circ})}+1=3}[/latex]

Deve haver alguma saída mais sagaz, mas eu realmente não a enxerguei. A propósito, da quarta linha da resolução para baixo eu fui aplicando Prostaférese (ou Fórmulas de Werner, como também são chamadas).