Question

How do you rewrite the expression as a single logarithm and simplify `ln(cos^2t)+ln(1+tan^2t)`?

Answer 1

`ln(cos^2(t))+ln(1+tan^2(t)) = color(green)0`

Explanation:

Things to remember:

[1]`color(white)("XXX")color(red)(tan=sin/cos)`

[2]`color(white)("XXX")color(red)(ln(a)+ln(b)=ln(a * b)`

[3]`color(white)("XXX")color(red)(cos^2+sin^2=1`

[4]`color(white)("XXX")color(red)(ln(a)=k if e^k=a)`

Therefore
`ln(cos^2(t))+ln(1+tan^2(t))`

`color(white)("XXX")=ln(cos^2(t)+sin^2(t))`

`color(white)("XXX")=ln(1)`

`color(white)("XXX")=0`

Answer 2

`0`.

Explanation:

We have to use the Rule `: ln a+ln b=ln(ab)`, together with the

Identity `: 1+tan^2t=sec^2t`

The Exp. `=lncos^2t+ln(1+tan^2t)`

`=lncos^2t+lnsec^2t`

`=ln{(cos^2t)(sec^2t)}`

`=ln 1`

`=0`.