Question

$\frac{cos^2\alpha+cos\alpha}{2cos^2\frac{\alpha}{2}}+1$

Answer 1

$\frac{cos \alpha (cos \alpha +1)}{1+cos \alpha } +1=cos \alpha +1$=2cos²$\frac{ \alpha }{2}$

Answer 2

$\frac{cos^2 \alpha +cos \alpha }{2cos^2\frac{ \alpha }{2}}+1=\frac{cos \alpha (cos \alpha +1)}{1+cos \alpha }+1=cos \alpha +1=2cos^2\frac{ \alpha }{2}\\\\P.S.\; \; \; cos \alpha +1=(cos^2\frac{ \alpha }{2}-sin^2\frac{ \alpha }{2})+(sin^2\frac{ \alpha }{2}+cos^2\frac{ \alpha }{2})=2cos^2\frac{ \alpha }{2}$