5.7. Упростите выражение $$\frac{\tan \alpha}{\cot \alpha} + \tan \beta \cdot \cot \beta$$;

Заменим $$\cot \alpha$$ на $$\frac{1}{\tan \alpha}$$ и $$\cot \beta$$ на $$\frac{1}{\tan \beta}$$: $$\frac{\tan \alpha}{\frac{1}{\tan \alpha}} + \tan \beta \cdot \frac{1}{\tan \beta} = \tan^2 \alpha + 1 = \frac{\sin^2 \alpha}{\cos^2 \alpha} + 1 = \frac{\sin^2 \alpha + \cos^2 \alpha}{\cos^2 \alpha} = \frac{1}{\cos^2 \alpha}$$. Итоговый ответ: **$$\frac{1}{\cos^2 \alpha}$$**