# How do you simplify the expression cost/(1+sint)+cost/(1-sint)?

Aug 27, 2016

$2 \sec t$

#### Explanation:

Begin by expressing the sum of the 2 fractions as a single fraction. This requires having a common denominator.

$\Rightarrow \frac{\cos t}{1 + \sin t} \times \frac{1 - \sin t}{1 - \sin t} + \frac{\cos t}{1 - \sin t} \times \frac{1 + \sin t}{1 + \sin t}$

$= \frac{\cos t \left(1 - \sin t\right) + \cos t \left(1 + \sin t\right)}{\left(1 + \sin t\right) \left(1 - \sin t\right)}$

distributing the brackets on numerator and denominator

$= \frac{\cos t - \cos t \sin t + \cos t + \cos t \sin t}{1 - {\sin}^{2} t}$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{{\sin}^{2} t + {\cos}^{2} t = 1 \Rightarrow 1 - {\sin}^{2} t = {\cos}^{2} t} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

simplifying numerator/denominator gives.

$\frac{2 \cos t}{{\cos}^{2} t} = \frac{2 {\cancel{\cos t}}^{1}}{{\cancel{\cos t}}^{1} \cos t} = \frac{2}{\cos} t = 2 \sec t$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\sec t = \frac{1}{\cos} t} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

$\Rightarrow \frac{\cos t}{1 + \sin t} + \frac{\cos t}{1 - \sin t} = 2 \sec t$