Айниятро исбот намоед:
\(\operatorname{tg}{\alpha}+\operatorname{ctg}{\alpha}+\operatorname{tg}{3\alpha}+\operatorname{ctg}{3\alpha}=\frac{8\cos^2{2\alpha}}{\sin{6\alpha}}\)
\(\operatorname{tg}{\alpha}+\operatorname{ctg}{\alpha}+\operatorname{tg}{3\alpha}+\operatorname{ctg}{3\alpha}=\)
\(=\frac{\sin{\alpha}}{\cos{\alpha}}+\frac{\cos{\alpha}}{\sin{\alpha}}+\frac{\sin{3\alpha}}{\cos{3\alpha}}+\frac{\cos{3\alpha}}{\sin3\alpha}=\)
\(=\frac{\sin^2{\alpha}+\cos^2{\alpha}}{\sin{\alpha}\cos{\alpha}}+\frac{\sin^2{3\alpha}+\cos^2{3\alpha}}{\sin{3\alpha}\cos{3\alpha}}=\)
\(=\frac{1}{\sin{\alpha}\cos{\alpha}}+\frac{1}{\sin{3\alpha}\cos{3\alpha}}=\)
\(=\frac{2}{2\sin{\alpha}\cos{\alpha}}+\frac{2}{2\sin{3\alpha}\cos{3\alpha}}\)
\(2\sin{\alpha}\cos{\alpha}=\sin{2\alpha}\)
\(2\sin{3\alpha}\cos{3\alpha}=\sin{6\alpha}\)
\(\operatorname{tg}{\alpha}+\operatorname{ctg}{\alpha}+\operatorname{tg}{3\alpha}+\operatorname{ctg}{3\alpha}=\)
\(=\frac{2}{\sin{2\alpha}}+\frac{2}{\sin{6\alpha}}=\frac{2(\sin{6\alpha}+\sin{2\alpha})}{\sin{2\alpha}\sin{6\alpha}}\)
\(\sin{6\alpha}+\sin{2\alpha}=2\sin{\frac{6\alpha+2\alpha}{2}}\cos{\frac{6\alpha-2\alpha}{2}}=2\sin{4\alpha}\cos{2\alpha}\)
\(\operatorname{tg}{\alpha}+\operatorname{ctg}{\alpha}+\operatorname{tg}{3\alpha}+\operatorname{ctg}{3\alpha}=\)
\(=\frac{2\cdot2\sin{4\alpha}\cos{2\alpha}}{\sin{2\alpha}\sin{6\alpha}}=\frac{4\sin{4\alpha}\cos{2\alpha}}{\sin{2\alpha}\sin{6\alpha}}\)
\(sin{4\alpha}=2\sin{2\alpha}\cos{2\alpha}\)
\(\operatorname{tg}{\alpha}+\operatorname{ctg}{\alpha}+\operatorname{tg}{3\alpha}+\operatorname{ctg}{3\alpha}=\)
\(=\frac{4\cdot2\sin{2\alpha}\cos{2\alpha}\cos{2\alpha}}{\sin{2\alpha}\sin{6\alpha}}=\frac{8\sin{2\alpha}\cos^2{2\alpha}}{\sin{2\alpha}\sin{6\alpha}}=\)
\(=\frac{8\cos^2{2\alpha}}{\sin{6\alpha}}\)
\(\operatorname{tg}{\alpha}+\operatorname{ctg}{\alpha}+\operatorname{tg}{3\alpha}+\operatorname{ctg}{3\alpha}=\frac{8\cos^2{2\alpha}}{\sin{6\alpha}}\)
Айният исбот шуд.