# How do you simplify the expression (tant+1)/sect?

Mar 6, 2018

$\sin t + \cos t$

#### Explanation:

Starting with the beginning expression, we replace $\tan t$ with $\sin \frac{t}{\cos} t$ and $\sec t$ with $\frac{1}{\cos} t$

$\frac{\tan t + 1}{\sec} t$ = $\frac{\sin \frac{t}{\cos} t + 1}{\frac{1}{\cos} t}$

Getting a common denominator in the numerator and adding,
$\textcolor{w h i t e}{a a a a a a a a}$=$\frac{\sin \frac{t}{\cos} t + \cos \frac{t}{\cos} t}{\frac{1}{\cos} t}$

$\textcolor{w h i t e}{a a a a a a a a}$= $\frac{\frac{\sin t + \cos t}{\cos} t}{\frac{1}{\cos} t}$
Dividing the numerator by the denominator,
$\textcolor{w h i t e}{a a a a a a a a}$=$\frac{\sin t + \cos t}{\cos} t \div \left(\frac{1}{\cos} t\right)$
Changing the divide to a multiply and inverting the fraction,
$\textcolor{w h i t e}{a a a a a a a a}$=$\frac{\sin t + \cos t}{\cos} t \times \left(\cos \frac{t}{1}\right)$
We see the $\cos t$ cancels out, leaving the resulting simplified expression.
$\textcolor{w h i t e}{a a a a a a a a}$=$\frac{\sin t + \cos t}{\cancel{\cos t}} \times \left(\frac{\cancel{\cos t}}{1}\right)$
$\textcolor{w h i t e}{a a a a a a a a}$=$\left(\sin t + \cos t\right)$