$a)$  

$Teza:\left|\cos \left(\frac{\alpha }{2}\right)\right|=\sqrt{\frac{1+\cos \alpha }{2}}$  

 

$P=\sqrt{\frac{1+\cos \alpha }{2}}=\sqrt{\frac{1+\cos \left(\frac{\alpha }{2}+\frac{\alpha }{2}\right)}{2}}=\sqrt{\frac{1+{\cos }^2\left(\frac{\alpha }{2}\right)-{\sin }^2\left(\frac{\alpha }{2}\right)}{2}}=$ $P=\sqrt{\frac{1+\cos \alpha }{2}}=\sqrt{\frac{1+\cos \left(\frac{\alpha }{2}+\frac{\alpha }{2}\right)}{2}}=\sqrt{\frac{1+{\cos }^2\left(\frac{\alpha }{2}\right)-{\sin }^2\left(\frac{\alpha }{2}\right)}{2}}=$  

$=\sqrt{\frac{{\sin }^2\left(\frac{\alpha }{2}\right)+{\cos }^2\left(\frac{\alpha }{2}\right)+{\cos }^2\left(\frac{\alpha }{2}\right)-{\sin }^2\left(\frac{\alpha }{2}\right)}{2}}=\sqrt{\frac{2{\cos }^2\left(\frac{\alpha }{2}\right)}{2}}=$  

$=\sqrt{{\cos }^2\left(\frac{\alpha }{2}\right)}=\left|\cos \left(\frac{\alpha }{2}\right)\right|$  

$L=P$  

 

$b)$  

$Teza:\left|\sin \left(\frac{\alpha }{2}\right)\right|=\sqrt{\frac{1-\cos \alpha }{2}}$  

 

$P=\sqrt{\frac{1-\cos \alpha }{2}}=\sqrt{\frac{1-\cos \left(\frac{\alpha }{2}+\frac{\alpha }{2}\right)}{2}}=\sqrt{\frac{1-{\cos }^2\left(\frac{\alpha }{2}\right)+{\sin }^2\left(\frac{\alpha }{2}\right)}{2}}=$  

$=\sqrt{\frac{{\sin }^2\left(\frac{\alpha }{2}\right)+{\cos }^2\left(\frac{\alpha }{2}\right)-{\cos }^2\left(\frac{\alpha }{2}\right)+{\sin }^2\left(\frac{\alpha }{2}\right)}{2}}=\sqrt{\frac{2{\sin }^2\left(\frac{\alpha }{2}\right)}{2}}=$  

 

$=\sqrt{{\sin }^2\left(\frac{\alpha }{2}\right)}=\left|\sin \left(\frac{\alpha }{2}\right)\right|$   

$L=P$