Question

تساوی زیر را اثبات کنید : با تشکر $$\frac {1}{\cos 0^{\circ} \cdot \cos 1^{\circ}} + \ldots +\frac {1}{\cos 88^{\circ} \cdot \cos 89^{\circ}} = \frac{\cos 1^{\circ}}{\sin 1^{\circ}}$$

Answer 1

$$ \frac{1}{\cos k \cos (k+1)}=\frac{1}{\sin 1}\frac{\sin 1}{\cos k \cos (k+1)}=\frac{1}{\sin 1} \frac{\sin(k+1)\cos(k)-\sin(k)\cos(k+1)}{\cos k \cos (k+1)} \\ =\frac{1}{\sin 1} \left(\tan(k+1)-\tan (k)\right) $$

$$ \sin 1\; \sum_{k=0}^{88}\frac{1}{\cos k \cos (k+1)}=\tan(89)-\tan(0)=\frac{\cos(1)}{\sin(1)} $$