Good evening:
Let's start from the left side and simplify:
[tex]\mathsf{\dfrac{cos(x)}{2(1+sin(x))}+\dfrac{cos(x)}{2(1-sin(x))}} = \mathsf{\dfrac{cos(x)(1-sin(x))+cos(x)(1+sin(x))}{2(1+sin(x))(1-sin(x))}}=\\ \\ \\ \mathsf{=\dfrac{\not{2}cos(x)}{\not2(1 - sin^2x)}} = \mathsf{\dfrac{cos(x)}{cos^2(x)}= \dfrac{1}{cos(x) }= \boxed{\mathsf{sec(x)}}}[/tex]
