oblicz bez użycia kalkulatora

[piwne_oko]

oblicz bez użycia kalkulatora: \(\displaystyle{ cos ^{4}105 -sin ^{4} 105}\)

[arpa007]

\(\displaystyle{ cos ^{4}105 -sin ^{4} 105\cos ^{4}105- cos^{4}15\(cos^{2}105)^2-(cos^{2}15)^2\(cos^{2}105+cos^{2}15)(cos^{2}105-cos^{2}105)

[(cos 15+cos 105)^{2}-2cos 15 cos 105] [(cos 15+cos 105)(cos15-cos 105)]

[(cos 15-sin 15)^{2}-2 cos15 - sin 15] [(cos 15-sin 15) (cos 15+sin 15)}\)

a wartosci sinus 15 i cosinus 15 juz latwo policzyc ;]

[wlodzimierz]

najprosciej:

\(\displaystyle{ cos^{4}105 - sin^{4}105 = (cos^{2}105 + sin^{2}105)(cos^{2}105 - sin^{2}105) = 1 - 2sin^{2}105 = cos210 = -cos30 = \frac{-\sqrt{3}}{2}}\)

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