Айниятро исбот намоед:

\(\frac{\operatorname{tg}{\alpha}-\cos^{-1}{\alpha}}{\cos{\alpha}-\operatorname{ctg}{\alpha}}=\operatorname{tg}{\alpha}\cdot\cos^{-1}{\alpha}\)

A \(= \frac{\operatorname{tg}{\alpha}-\cos^{-1}{\alpha}}{\cos{\alpha}-\operatorname{ctg}{\alpha}}\)

\(\operatorname{tg}{x}=\frac{\sin{x}}{\cos{x}}\)

\(\operatorname{ctg}{x}=\frac{\cos{x}}{\sin{x}}\)

A\(=\frac{\frac{\sin{\alpha}}{\cos{\alpha}}-\frac{1}{\cos{\alpha}}}{\cos{\alpha}-\frac{\cos{\alpha}}{\sin{\alpha}}}=\)

\(=\frac{\frac{1}{\cos{\alpha}}(\sin{\alpha}-1)}{\frac{\cos{\alpha}}{\sin{\alpha}}(\sin{\alpha}-1)}=\)

\(=\frac{\frac{1}{\cos{\alpha}}}{\frac{\cos{\alpha}}{\sin{\alpha}}}=\frac{1}{\cos{\alpha}}:\frac{\cos{\alpha}}{\sin{\alpha}}=\)

\(=\frac{1}{\cos{\alpha}}\cdot\frac{\sin{\alpha}}{\cos{\alpha}}=\frac{1}{\cos{\alpha}}\cdot\operatorname{tg}{\alpha}=\)

\(=\operatorname{tg}{\alpha}\cdot\cos^{-1}{\alpha}\)

\(\frac{\operatorname{tg}{\alpha}-\cos^{-1}{\alpha}}{\cos{\alpha}-\operatorname{ctg}{\alpha}}=\operatorname{tg}{\alpha}\cdot\cos^{-1}{\alpha}\)

Айният исбот шуд.