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Problema \n<\/div>\n \n Solu\u00e7\u00e3o<\/p>\n Solu\u00e7\u00e3o elaborada pelos Moderadores do Blog<\/strong>.<\/p>\n<\/div>\n \n","protected":false},"excerpt":{"rendered":" Problema (Indicado a partir do 1\u00ba ano do E. M.) Simplifique a express\u00e3o [tex]\\sqrt{sen^4 x + 4 cos^2 x}-\\sqrt{cos^4 x + 4 sen^2 x}[\/tex] Solu\u00e7\u00e3o Observe que: [tex]\\qquad \\sqrt{sen^4 x + 4 cos^2 x}-\\sqrt{cos^4 x + 4 sen^2 x}=[\/tex] [tex]\\qquad = \\sqrt{sen^4 x + 4 (1 – sen^2 x)}-\\sqrt{cos^4 x + 4 (1-cos^2 x)}[\/tex] [tex]\\qquad … <\/p>\n
\n (Indicado a partir do 1\u00ba ano do E. M.)<\/span><\/p>\n
\nSimplifique a express\u00e3o [tex]\\sqrt{sen^4 x + 4 cos^2 x}-\\sqrt{cos^4 x + 4 sen^2 x}[\/tex]<\/p>\n
\nObserve que:
\n[tex]\\qquad \\sqrt{sen^4 x + 4 cos^2 x}-\\sqrt{cos^4 x + 4 sen^2 x}=[\/tex]
\n[tex]\\qquad = \\sqrt{sen^4 x + 4 (1 – sen^2 x)}-\\sqrt{cos^4 x + 4 (1-cos^2 x)}[\/tex]
\n[tex]\\qquad = \\sqrt{sen^4 x – 4 sen^2 x + 4}-\\sqrt{cos^4 x-4 cos^2 x + 4}[\/tex]
\n[tex]\\qquad = \\sqrt{(2-sen^2 x)^2}-\\sqrt{(2- cos^2 x)^2}[\/tex]
\n[tex]\\qquad = |2-sen^2 x|-|2-cos^2 x|[\/tex].
\nComo [tex]-1 \\leq senx \\, , \\, cosx \\leq 1[\/tex], ent\u00e3o [tex]0 \\leq sen^2x \\, , \\, cos^2x \\leq 1 < 2[\/tex], donde [tex]0 < 2-sen^2x \\, \\, [\/tex] e [tex] \\, \\, 0 < 2-cos^2x[\/tex].
\n Com isso, [tex]|2 – sen^2 x|=2 – sen^2 x \\, \\, [\/tex] e [tex] \\, \\, |2 – cos^2 x|=2 – cos^2 x \\, \\, [\/tex] e, ent\u00e3o,
\n[tex]\\qquad \\sqrt{sen^4 x + 4 cos^2 x}-\\sqrt{cos^4 x + 4 sen^2 x}= 2-sen^2 x-2 + cos^2 x = cos(2x)[\/tex].<\/p>\n
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