Айниятро исбот кунед:
\((\cos^{-1}{2\alpha}+\operatorname{ctg}({\frac{5\pi}{2}+2\alpha}))\operatorname{ctg}({\frac{5\pi}{4}-\alpha}) = 1\)
\(A = (\cos^{-1}{2\alpha}+\operatorname{ctg}({\frac{5\pi}{2}+2\alpha}))\operatorname{ctg}({\frac{5\pi}{4}-\alpha})\)
\(\cos^{-1}{2\alpha} = \frac{1}{\cos{2\alpha}}\)
\(\operatorname{ctg}({\frac{5\pi}{2}+2\alpha}) = \operatorname{ctg}({2\pi+\frac{\pi}{2}+2\alpha}) = \operatorname{ctg}({\frac{\pi}{2}+2\alpha}) =\)
\(= -\operatorname{tg}{2\alpha} =-\frac{\sin{2\alpha}}{\cos{2\alpha}}\)
\(A = \left(\frac{1}{\cos{2\alpha}}+(-\frac{\sin{2\alpha}}{\cos{2\alpha}})\right)\operatorname{ctg}({\frac{5\pi}{4}-\alpha})\)
\(\operatorname{ctg}({\frac{5\pi}{4}-\alpha}) = \operatorname{ctg}({\pi+\frac{\pi}{4}-\alpha}) = \operatorname{ctg}({\frac{\pi}{4}-\alpha}) = \)
\( = \frac{\cos{\frac{\pi}{4}-\alpha}}{\sin{\frac{\pi}{4}-\alpha}} = \frac{\cos{\frac{\pi}{4}}cos{\alpha}+\sin{\frac{\pi}{4}}sin{\alpha}}{\sin{\frac{\pi}{4}}cos{\alpha}-\cos{\frac{\pi}{4}}sin{\alpha}} = \frac{\cos{\alpha}+\sin{\alpha}}{\cos{\alpha}-\sin{\alpha}}\)
\(A = (\frac{1}{\cos{2\alpha}}-\frac{\sin{2\alpha}}{\cos{2\alpha}})\cdot\frac{\cos{\alpha}+\sin{\alpha}}{\cos{\alpha}-sin{\alpha}}\)
\(\frac{1}{\cos{2\alpha}}-\frac{\sin{2\alpha}}{\cos{2\alpha}} = \frac{1-\sin{2\alpha}}{\cos{2\alpha}} = \frac{1-2\sin{\alpha}\cos{\alpha}}{\cos{2\alpha}}=\)
\( = \frac{\cos^2{\alpha}-2\sin{\alpha}\cos{\alpha}+\sin^2{\alpha}}{\cos{2\alpha}}\)
\(\frac{\cos^2{\alpha}-2\sin{\alpha}\cos{\alpha}+\sin^2{\alpha}}{\cos{2\alpha}} = \frac{\cos^2{\alpha}-2\sin{\alpha}\cos{\alpha}+\sin^2{\alpha}}{\cos^2{\alpha}-\sin^2{\alpha}} =\)
\(= \frac{(\cos{\alpha}-\sin{\alpha})(\cos{\alpha}-\sin{\alpha})}{(\cos{\alpha}+\sin{\alpha})(\cos{\alpha}-\sin{\alpha})}=\)
\(= \frac{\cos{\alpha}-\sin{\alpha}}{\cos{\alpha}+\sin{\alpha}}\)
\(A = \frac{\cos{\alpha}-\sin{\alpha}}{\cos{\alpha}+\sin{\alpha}}\cdot\frac{\cos{\alpha}+\sin{\alpha}}{\cos{\alpha}-\sin{\alpha}} = 1\)
\(A = 1\)
Исбот шуд.