Question

ПОМОГИТЕ С РЕШЕНИЕМ!

$\frac{cos20 - cos140}{cos20 + cos140} =$

$\frac{cos25 - cos85}{sin25 + sin85} =$
Это всё в градусах!

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Answer 1

$\dfrac{\cos20^\circ-\cos140^\circ}{\cos20^\circ+\cos140^\circ}=\dfrac{-2\sin\frac{20^\circ+140^\circ}{2}\sin\frac{20^\circ-140^\circ}{2} }{2\cos\frac{20^\circ+140^\circ}{2}\cos\frac{20^\circ-140^\circ}{2} }=-\mathrm{tg}\frac{20^\circ+140^\circ}{2}\mathrm{tg}\frac{20^\circ-140^\circ}{2}=\\=-\mathrm{tg}80^\circ\mathrm{tg}(-60^\circ)=\mathrm{tg}80^\circ\mathrm{tg}60^\circ=\mathrm{tg}80^\circ\cdot\sqrt{3} =\sqrt{3}\cdot\mathrm{tg}80^\circ$

$\dfrac{\cos25^\circ-\cos85^\circ}{\sin25^\circ+\sin85^\circ}=\dfrac{-2\sin\frac{25^\circ+85^\circ}{2}\sin\frac{25^\circ-85^\circ}{2} }{2\sin\frac{25^\circ+85^\circ}{2}\cos\frac{25^\circ-85^\circ}{2} }=-\mathrm{tg}\frac{25^\circ-85^\circ}{2}=\\=-\mathrm{tg}(-30^\circ)=\mathrm{tg}30^\circ=\dfrac{\sqrt{3}}{3}$