# What is (1+cosx)/(1+secx) for x=pi/3?

Jan 2, 2017

$\frac{1 + \cos x}{1 + \sec x} = \cos x$ and for $x = \frac{\pi}{3}$, it is $\frac{1}{2}$.

#### Explanation:

$\frac{1 + \cos x}{1 + \sec x}$

= $\frac{1 + \cos x}{1 + \frac{1}{\cos} x}$

= $\frac{1 + \cos x}{\frac{\cos x + 1}{\cos} x}$

= $\left(1 + \cos x\right) \times \cos \frac{x}{1 + \cos x}$

= $\cos x$

Hence $\frac{1 + \cos x}{1 + \sec x} = \cos x$ for all values of $x$

and for $x = \frac{\pi}{3}$

$\frac{1 + \cos x}{1 + \sec x} = \frac{1 + \cos \left(\frac{\pi}{3}\right)}{1 + \sec \left(\frac{\pi}{3}\right)}$

= $\frac{1 + \frac{1}{2}}{1 + 2} = \frac{\frac{3}{2}}{3} = \frac{3}{2} \times \frac{1}{3} = \frac{1}{2}$

As $\cos x = \cos \left(\frac{\pi}{3}\right) = \frac{1}{2}$

for $x = \frac{\pi}{3}$, $\frac{1 + \cos x}{1 + \sec x} = \cos x$