Question

10√2 cos (−п/12) − 5√3 .

Answer 1

$cosfrac{pi}{12}=cos, 15^circ =sqrt{ frac{1+cos30^circ }{2} }=sqrt{ frac{1+frac{sqrt3}{2}}{2}} = sqrt{ frac{2+sqrt3}{4}} = frac{ sqrt{2+sqrt3} }{2} \\\10sqrt2cdot cos(-frac{pi}{12})-5sqrt3=10sqrt2cdot cos frac{pi}{12} -5sqrt3=\\=10sqrt2cdot frac{sqrt{2+sqrt3}}{2} -5sqrt3=5sqrt2cdot sqrt{2+sqrt3}-5sqrt3=\\=5cdot sqrt{2cdot (2+sqrt3)}-5sqrt3=5cdot sqrt{4+2sqrt3}-5sqrt3=\\=5cdot sqrt{(1+sqrt3)^2}-5sqrt3=5cdot (1+sqrt3)-5sqrt3=5+5sqrt3-5sqrt3=5$

$P.S.; ; cosfrac{pi}{12}= frac{sqrt{2+sqrt3}}{2} = frac{sqrt2cdot sqrt{2+sqrt3}}{2sqrt2}=frac{sqrt{2cdot (2+sqrt3)}}{2sqrt2} = \\=frac{sqrt{4+2sqrt3}}{2sqrt2}=frac{sqrt{(sqrt3+1)^2}}{2sqrt2} = frac{sqrt3+1}{2sqrt2}$