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

Mar 16, 2018

$\ln | \cot t | + \ln \left(1 + {\tan}^{2} t\right) \equiv \ln | 2 \csc \left(2 t\right) |$

#### Explanation:

We use the rules of logarithms:

$\log A B \equiv \log A + \log B$

So, we can write the expression as:

$\ln | \cot t | + \ln \left(1 + {\tan}^{2} t\right) \equiv \ln \left(| \cot t | \left(1 + {\tan}^{2} t\right)\right)$

$\text{ } = \ln \left(| \cot t \left(1 + {\tan}^{2} t\right) |\right)$

$\text{ } = \ln | \cos \frac{t}{\sin} t \frac{1}{\cos} ^ 2 t |$

$\text{ } = \ln | \frac{1}{\sin} t \frac{1}{\cos} t |$

$\text{ } = \ln | \frac{1}{\sin t \cos t} |$

$\text{ } = \ln | \frac{2}{2 \sin t \cos t} |$

$\text{ } = \ln | \frac{2}{\sin} \left(2 t\right) |$

$\text{ } = \ln | 2 \csc \left(2 t\right) |$