# How do you verify the identity sec^2(x/2)= (2secx + 2)/(secx + 2 + cosx)?

Jul 22, 2015

Required to prove : ${\sec}^{2} \left(\frac{x}{2}\right) = \frac{2 \sec x + 2}{\sec x + 2 + \cos x}$

$\text{Right Hand Side} = \frac{2 \sec x + 2}{\sec x + 2 + \cos x}$

Remember that $\sec x = \frac{1}{\cos} x$

$\implies \frac{2 \cdot \frac{1}{\cos} x + 2}{\frac{1}{\cos} x + 2 + \cos x}$

Now, multiply top and bottom by $\cos x$

$\implies \frac{\cos x \times \left(2 \cdot \frac{1}{\cos} x + 2\right)}{\cos x \times \left(\frac{1}{\cos} x + 2 + \cos x\right)}$

$\implies \frac{2 + 2 \cos x}{1 + 2 \cos x + {\cos}^{2} x}$

Factorize the bottom,

$\implies \frac{2 \left(1 + \cos x\right)}{1 + \cos x} ^ 2$

$\implies \frac{2}{1 + \cos x}$

Recall the identity : $\cos 2 x = 2 {\cos}^{2} x - 1$
$\implies 1 + \cos 2 x = 2 {\cos}^{2} x$

Similarly : $1 + \cos x = 2 {\cos}^{2} \left(\frac{x}{2}\right)$

$\implies \text{Right Hand Side"=2/(2cos^2(x/2))=1/cos^2(x/2)=color(blue)(sec^2(x/2)) = "Left Hand Side}$

As Required