question about trigonometric function transformation what is the difference between 
$\cos^2(\theta - 180) $ and  $\cos^2(180 - \theta) $ Does $\\cosec^2(450 + \theta)$ transform into $\sec$?

There is no difference between $\cos^2 (\theta-180)$ & $\cos^2 (\theta-180)$ since, $$ \cos (-\theta)=\cos (\theta)$$ $$\implies \cos^2 (\theta-180)=(\cos (\theta-180))^2$$$$=(\cos(-(180-\theta)))^2=(\cos (180-\theta))^2=\cos^2(180-\theta)$$ Notice, $$cosec^2(450+\theta)=cosec^2(360+90+\theta)$$ $$=cosec^2(90+\theta)$$ $$=sec^2(\theta)$$ 
