undersetx→2lim((cos α)^x+(sin α)^x-1)/(x-2) is equal to

$\underset{x→2}{\lim}\frac{(\cos α)^x+(\sin α)^x-1}{x-2}=\underset{x→2}{\lim}\frac{(\cos α)^x\log_e(\cos α)+(\sin α)^x\log_e(\sin α)}{1}$

Using L’Hospital rule

$=\cos^2α(\log_e\cos α)+\sin^2α\log_e(\sin α)$