Aloha :)
Mit Hilfe der Additionstheoreme für Sinus und Cosinus geht das wie folgt:
$$\left.\sin\alpha\stackrel{!}{=}\sin(\alpha-72^o)=\sin\alpha\cos72^o-\cos\alpha\sin72^o\quad\right|\;:\sin\alpha$$$$\left.1=\cos72^o-\cot\alpha\sin72^o\quad\right|\;+\cot\alpha\sin72^o$$$$\left.\cot\alpha\sin72^o+1=\cos72^o\quad\right|\;-1$$$$\left.\cot\alpha\sin72^o=\cos72^o-1\quad\right|\;:\sin72^o$$$$\left.\cot\alpha\sin72^o=\cos72^o-1\quad\right|\;:\sin72^o$$$$\left.\cot\alpha=\frac{\cos72^o-1}{\sin72^o}\quad\right.\;$$$$\alpha=-54^o\to126^o$$
$$\left.\cos\alpha\stackrel{!}{=}\cos(\alpha+16^o)=\cos\alpha\cos16^o-\sin\alpha\sin16^o\quad\right|\;:\cos\alpha$$$$\left.1=\cos16^o-\tan\alpha\sin16^o\quad\right|\;+\tan\alpha\sin16^o$$$$\left.\tan\alpha\sin16^o+1=\cos16^o\quad\right|\;-1$$$$\left.\tan\alpha\sin16^o=\cos16^o-1\quad\right|\;:\sin16^o$$$$\left.\tan\alpha=\frac{\cos16^o-1}{\sin16^o}\quad\right.$$$$\alpha=-8^o\to172^o$$