簡約 8ln(cos^2(t))+8ln(1+tan^2(t))

解

簡約

8 ln (cos^{2} (t)) + 8 ln (1 + tan^{2} (t))

ln (sec^{16} (t) cos^{16} (t))

解答ステップ

8 ln (cos^{2} (t)) + 8 ln (1 + tan^{2} (t))

対数の規則を適用する:

a lo g_{c} (b) = lo g_{c} (b^{a})

8 ln (cos^{2} (t)) = ln (cos^{2} (t)^{8})

= ln (cos^{2} (t)^{8}) + 8 ln (tan^{2} (t) + 1)

対数の規則を適用する:

a lo g_{c} (b) = lo g_{c} (b^{a})

8 ln (tan^{2} (t) + 1) = ln ((tan^{2} (t) + 1)^{8})

= ln (cos^{2} (t)^{8}) + ln ((tan^{2} (t) + 1)^{8})

対数の規則を適用する:

lo g_{c} (a) + lo g_{c} (b) = lo g_{c} (ab)

ln (cos^{2} (t)^{8}) + ln ((1 + tan^{2} (t))^{8}) = ln (cos^{2} (t)^{8} (tan^{2} (t) + 1)^{8})

= ln (cos^{2} (t)^{8} (tan^{2} (t) + 1)^{8})

簡素化

cos^{2} (t)^{8} (1 + tan^{2} (t))^{8} : sec^{16} (t) cos^{16} (t)

= ln (sec^{16} (t) cos^{16} (t))