Question

Найдите такое а, чтобы выполнялось тождество
$(1 + tg(a \cdot sin^2(\frac{17\pi}{32})))(1 + tg(a \cdot cos^2(\frac{17\pi}{32}))) = 2$

Answer 1

$\left [ 1+\mathrm{tg}\left ( a\sin^2 \cfrac{17\pi}{32} \right ) \right ]\left [ 1+\mathrm{tg}\left ( a\cos^2 \cfrac{17\pi}{32} \right ) \right ]=2\\1+\mathrm{tg}\left ( a\sin^2 \cfrac{17\pi}{32} \right )+\mathrm{tg}\left ( a\cos^2 \cfrac{17\pi}{32} \right )+\mathrm{tg}\left ( a\sin^2 \cfrac{17\pi}{32} \right )\mathrm{tg}\left ( a\cos^2 \cfrac{17\pi}{32} \right )=2$$\mathrm{tg}\left ( a\sin^2 \cfrac{17\pi}{32} \right )+\mathrm{tg}\left ( a\cos^2 \cfrac{17\pi}{32} \right )+\mathrm{tg}\left ( a\sin^2 \cfrac{17\pi}{32} \right )\mathrm{tg}\left ( a\cos^2 \cfrac{17\pi}{32} \right )=1$$\frac{\sin \left ( a\sin^2\cfrac{17\pi}{32} \right )}{\cos \left ( a\sin^2\cfrac{17\pi}{32} \right )}+\frac{\sin \left ( a\cos^2\cfrac{17\pi}{32} \right )}{\cos \left ( a\cos^2\cfrac{17\pi}{32} \right )}+\frac{\sin \left ( a\sin^2\cfrac{17\pi}{32} \right )\sin \left ( a\cos^2\cfrac{17\pi}{32} \right )}{\cos \left ( a\sin^2\cfrac{17\pi}{32} \right )\cos \left ( a\cos^2\cfrac{17\pi}{32} \right )}=1$$\sin \left ( a\sin^2\cfrac{17\pi}{32} \right )\cos \left ( a\cos^2\cfrac{17\pi}{32} \right )+\cos \left ( a\sin^2\cfrac{17\pi}{32} \right )\sin \left ( a\cos^2\cfrac{17\pi}{32} \right )+\\+\sin \left ( a\sin^2\cfrac{17\pi}{32} \right )\sin \left ( a\cos^2\cfrac{17\pi}{32} \right )=\cos \left ( a\sin^2\cfrac{17\pi}{32} \right )\cos \left ( a\cos^2\cfrac{17\pi}{32} \right )$$\sin \left ( a\sin^2 \cfrac{17\pi}{32}+a\cos^2\cfrac{17\pi}{32} \right )=\cos\left ( a\sin^2 \cfrac{17\pi}{32}+a\cos^2\cfrac{17\pi}{32} \right )\Rightarrow \\\Rightarrow \sin a=\cos a\Leftrightarrow \mathrm{tg} \; a=1\Rightarrow a=\cfrac{\pi}{4}$