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Айниятро исбот намоед:
\(\frac{\cos{(3\pi-2\alpha)}}{2\sin^2{(\frac{5\pi}{4}+\alpha)}} = \operatorname{tg}{(\alpha-\frac{5\pi}{4})}\)
\(\cos{(3\pi-2\alpha)} = \cos{(2\pi+\pi-2\alpha)} = \cos{(\pi-2\alpha)}\)
\(\frac{\cos{(\pi-2\alpha)}}{2\sin^2{(\frac{5\pi}{4}+\alpha)}} = \frac{-\cos{2\alpha}}{2(\sin{\frac{5\pi}{4}}\cos{\alpha}+\sin{\alpha}\cos{\frac{5\pi}{4}})^2}\)
\((\sin{\frac{5\pi}{4}}\cos{\alpha}+\sin{\alpha}\cos{\frac{5\pi}{4}})^2 = (-\frac{\sqrt{2}}{2}\cos{\alpha}+(-\frac{\sqrt{2}}{2})\sin{\alpha})^2 \)
\(= (-\frac{\sqrt{2}}{2}(\cos{\alpha}+\sin{\alpha}))^2 = \frac{2}{4}(\cos{\alpha}+\sin{\alpha})^2\)
\(\frac{-\cos{2\alpha}}{2(\sin{\frac{5\pi}{4}}\cos{\alpha}+\sin{\alpha}\cos{\frac{5\pi}{4}})^2} = \frac{-\cos{2\alpha}}{2\cdot\frac{2}{4}(\cos{\alpha}+\sin{\alpha})^2} \)
\(= \frac{-(\cos^2{\alpha}-\sin^2{\alpha})}{(\cos{\alpha}+\sin{\alpha})^2} = \frac{-(\cos{\alpha}-\sin{\alpha})}{\cos{\alpha}+\sin{\alpha}} = \frac{\sin{\alpha}-\cos{\alpha}}{\cos{\alpha}+\sin{\alpha}}\)
\(\frac{\cos{(3\pi-2\alpha)}}{2\sin^2{(\frac{5\pi}{4}+\alpha)}} = \frac{\sin{\alpha}-\cos{\alpha}}{\cos{\alpha}+\sin{\alpha}}(*)\)
\( \operatorname{tg}{(\alpha-\frac{5\pi}{4})} = \operatorname{tg}{(\alpha-\frac{\pi}{4})} = \frac{\sin(\alpha-\frac{\pi}{4})}{\cos(\alpha-\frac{\pi}{4})} =\)
\(= \frac{\sin\alpha\cos{\frac{\pi}{4}}-\cos\alpha\sin{\frac{\pi}{4}}}{\cos\alpha\cos{\frac{\pi}{4}}+\sin\alpha\sin{\frac{\pi}{4}}} =\)
\(= \frac{\frac{\sqrt{2}}{2}(\sin{\alpha}-\cos{\alpha})}{\frac{\sqrt{2}}{2}(\sin{\alpha}+\cos{\alpha})} = \frac{\sin{\alpha}-\cos{\alpha}}{\cos{\alpha}+\sin{\alpha}}\)
\( \operatorname{tg}{(\alpha-\frac{5\pi}{4})} = \frac{\sin{\alpha}-\cos{\alpha}}{\cos{\alpha}+\sin{\alpha}}(**)\)
Аз (*) ва (**) мебарояд, ки
\(\frac{\cos{(3\pi-2\alpha)}}{2\sin^2{(\frac{5\pi}{4}+\alpha)}} = \operatorname{tg}{(\alpha-\frac{5\pi}{4}})\).
Айният исбот шуд!