Explicație pas cu pas:
Prelucram membrul stang si folosim formulele:
- [tex] ctg~x=\frac{cos~x}{sin~x} [/tex]
- [tex] sin~(x-y)=sin~x*cos~y-sin~y*cos~x [/tex]
- [tex] sin~x*sin~y=\frac{1}{2}*[cos~(x-y)-cos~(x+y)] [/tex]
- [tex] cos~90^{\circ}=0 [/tex]
- [tex] cos~(-x)=cos~x [/tex]
[tex](ctg10^{\circ}-ctg80^{\circ})*cos70^{\circ}=\\=(\frac{cos10^{\circ}}{sin10^{\circ}}-\frac{cos80^{\circ}}{sin80^{\circ}})*cos70^{\circ}=\\=(\frac{cos10^{\circ}*sin80^{\circ}}{sin10^{\circ}*sin80^{\circ}}-\frac{cos80^{\circ}*sin10^{\circ}}{sin10^{\circ}*sin80^{\circ}})*cos70^{\circ}=\\=\frac{sin80^{\circ}*cos10^{\circ}-cos80^{\circ}*sin10^{\circ}}{sin10^{\circ}*sin80^{\circ}}*cos70^{\circ}=\\=\frac{sin(80^{\circ}-10^{\circ})}{sin10^{\circ}*sin80^{\circ}}*cos70^{\circ}=\\[/tex]
[tex]=\frac{sin70^{\circ}*cos70^{\circ}}{sin10^{\circ}*sin80^{\circ}}=\\=\frac{sin70^{\circ}*cos70^{\circ}}{\frac{1}{2}*[cos(10^{\circ}-80^{\circ})-cos(10^{\circ}+80^{\circ})]}=\\=\frac{2*sin70^{\circ}*cos70^{\circ}}{cos(-70^{\circ})-cos90^{\circ}}=\\=\frac{2*sin70^{\circ}*cos70^{\circ}}{cos70^{\circ}-0}=\\=\frac{2*sin70^{\circ}*cos70^{\circ}}{cos70^{\circ}}=\\=2sin70^{\circ}~(relatia~1)[/tex]
Prelucram membrul drept si folosim formula:
- [tex] sin~(180^{\circ}-x)=sin~x [/tex]
[tex]2*sin110^{\circ}=2*sin(180^{\circ}-110^{\circ})=2*sin70^{\circ}~(relatia~2)[/tex]
Obervam ca relatia 1 coincide cu relatia 2 si, deci, are loc egalitatea:
[tex](ctg10^{\circ}-ctg80^{\circ})*cos70^{\circ}=2*sin110^{\circ}[/tex]
