# How do you verify the identity sec(u/2)=+-sqrt((2tanu)/(tanu+sinu))?

Feb 18, 2017

Square both sides of the equation
$\frac{2 \tan u}{\tan u + \sin u} = {\sec}^{2} \left(\frac{u}{2}\right)$
Develop the left side:
$L S = \frac{\frac{2 \sin u}{\cos u}}{\sin \frac{u}{\cos u} + \sin u} =$

$= \frac{2 \sin u}{\sin u + \sin u . \cos u} = \frac{2 \sin u}{\left(\sin u\right) \left(1 + \cos u\right)}$

Replace $\left(1 + \cos u\right)$ by $2 {\cos}^{2} \left(\frac{u}{2}\right)$, we get:

$L S = \frac{1}{\cos} ^ 2 \left(\frac{u}{2}\right) = {\sec}^{2} \left(\frac{u}{2}\right)$