Let's apply the cos of a sum formula to "\\cos(15^\\circ+15^\\circ)", as we know that "\\cos(30^{\\circ})=\\frac{\\sqrt{3}}{2}:"
"\\frac{\\sqrt{3}}{2} = \\cos(30^\\circ)=\\cos(15^\\circ+15^\\circ)=\\cos(15^\\circ)\\times\\cos(15^\\circ)-\\sin(15^\\circ)\\times\\sin(15^\\circ)"
Now by the main trigonometrical identity ("\\sin^2\\alpha+\\cos^2\\alpha=1)" we can write :
"\\frac{\\sqrt{3}}{2}=\\cos^2(15^\\circ)-(1-\\cos^2(15^\\circ))=2\\cos^2(15^\\circ)-1"
"2\\cos^2(15^\\circ)=1+\\frac{\\sqrt{3}}{2}"
"\\cos(15^\\circ)=\\pm \\sqrt{\\frac{1}{2}+\\frac{\\sqrt{3}}{4}}"
We know also that "\\cos(15^\\circ)>0" (as "0^\\circ<15^\\circ<90^\\circ" ), so we have :
"\\cos(15^\\circ)=\\sqrt{\\frac{1}{2}+\\frac{\\sqrt{3}}{4}}=\\frac{1}{2} \\sqrt{2+\\sqrt{3}}"
We can also simplify the last expression by writing :
"(x+y)^2=2+\\sqrt{3}"
"x^2+y^2=2, xy=\\frac{\\sqrt{3}}{2}"
"x^2+\\frac{3}{4x^2}=2"
"x^4-2x^2+\\frac{3}{4}=0"
"x^2=\\frac{2\\pm\\sqrt{4-3}}{2}"
"x=\\sqrt{\\frac{1}{2}},y=\\sqrt{\\frac{3}{2}}"
So we have :
"\\cos(15^\\circ)=\\frac{1}{2}\\times(\\frac{1}{\\sqrt{2}}+\\frac{\\sqrt{3}}{\\sqrt{2}})=\\frac{\\sqrt{2}+\\sqrt{6}}{4}"
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