How do you simplify (1+cos y)/(1+sec y)?

Aug 5, 2015

$\frac{1 + \cos y}{1 + \sec y} = \cos y$

Explanation:

$\sec y = \frac{1}{\cos} y$, therefore we have:

$\frac{1 + \cos y}{1 + \sec y} = \left(\cos \frac{y}{\cos} y\right) \left(\frac{1 + \cos y}{1 + \frac{1}{\cos} y}\right) =$
$\cos y \left(\frac{1 + \cos y}{1 + \cos y}\right) = \cos y$

Aug 6, 2015

Since $\sec y = \frac{1}{\cos} y$, you can rewrite this as:

$\frac{1 + \cos y}{1 + \left(\frac{1}{\cos} y\right)}$

Thus, multiply through by $\left(\frac{\cos y}{\cos y}\right)$:

$= \frac{\cos y + {\cos}^{2} y}{\cos y + 1}$

$= \cos y \left(\cancel{\frac{1 + \cos y}{\cos y + 1}}\right)$

$= \textcolor{b l u e}{\cos y}$