What is 5ln(cos^2(t))+5ln(1+tan^2(t)) ?

The solution to 5ln(cos^2(t))+5ln(1+tan^2(t)) is ln(sec^{10}(t)cos^{10}(t))

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simplify 5ln(cos^2(t))+5ln(1+tan^2(t))

simplify

5 ln (cos^{2} (t)) + 5 ln (1 + tan^{2} (t))

ln (sec^{10} (t) cos^{10} (t))

Solution steps

5 ln (cos^{2} (t)) + 5 ln (1 + tan^{2} (t))

Apply log rule:

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

5 ln (cos^{2} (t)) = ln (cos^{2} (t)^{5})

= ln (cos^{2} (t)^{5}) + 5 ln (tan^{2} (t) + 1)

Apply log rule:

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

5 ln (tan^{2} (t) + 1) = ln ((tan^{2} (t) + 1)^{5})

= ln (cos^{2} (t)^{5}) + ln ((tan^{2} (t) + 1)^{5})

Apply log rule:

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

ln (cos^{2} (t)^{5}) + ln ((1 + tan^{2} (t))^{5}) = ln (cos^{2} (t)^{5} (tan^{2} (t) + 1)^{5})

= ln (cos^{2} (t)^{5} (tan^{2} (t) + 1)^{5})

Simplify

cos^{2} (t)^{5} (1 + tan^{2} (t))^{5} : sec^{10} (t) cos^{10} (t)

= ln (sec^{10} (t) cos^{10} (t))

\frac{3}{2} ln (3)

- lo g_{10} (2.5 \cdot 1 0^{- 7})

lo g_{10} (\frac{x z ^{2}}{y})

ln (\frac{x}{x ^{2} + 1})

ln (\frac{e x ^{2}}{y})

What is 5ln(cos^2(t))+5ln(1+tan^2(t)) ?
The solution to 5ln(cos^2(t))+5ln(1+tan^2(t)) is ln(sec^{10}(t)cos^{10}(t))