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Question Number 154892 by mathdanisur last updated on 22/Sep/21

$$\mathrm{Find}:$$$$\cos \left(\frac{3\pi }{7}\right)+\cos \left(\frac{3\pi }{7}\right)\sqrt{2-2\cos \left(\frac{3\pi }{7}\right)}=?$$

Commented by MJS_new last updated on 23/Sep/21

$$\frac{1}{2}$$

Commented by mathdanisur last updated on 23/Sep/21

$$\mathrm{Thank}\mathrm{you}\mathbf{S}\mathrm{er},\mathrm{solution}\mathrm{if}\mathrm{possible}$$

Answered by mnjuly1970 last updated on 23/Sep/21

$$cos\left(\frac{3\pi }{7}\right)+2cos\left(\frac{3\pi }{7}\right).sin\left(\frac{3\pi }{14}\right)$$$$cos\left(\frac{3\pi }{7}\right)+2cos\left(\frac{3\pi }{7}\right).cos(\frac{\pi }{2}-\frac{3\pi }{14})$$$$=cos\left(\frac{3\pi }{7}\right)+2cos\left(\frac{3\pi }{7}\right)cos\left(\frac{2\pi }{7}\right)$$$$\therefore \mathrm{A}:=cos\left(\frac{3\pi }{7}\right)+cos\left(\frac{\pi }{7}\right)+cos\left(\frac{5\pi }{7}\right)$$$$2\mathrm{A}.sin\left(\frac{\pi }{7}\right)=sin\left(\frac{4\pi }{7}\right)-sin\left(\frac{2\pi }{7}\right)$$$$+sin\left(\frac{2\pi }{7}\right)+sin\left(\frac{6\pi }{7}\right)-sin\left(\frac{4\pi }{7}\right)$$$$2\mathrm{A}.sin\left(\frac{\pi }{7}\right)=sin(\pi -\frac{\pi }{7})$$$$\mathrm{A}=\frac{1}{2}◼m.n$$$$$$

Commented by mathdanisur last updated on 23/Sep/21

$$\mathrm{Very}\mathrm{nice}\mathrm{solution},\mathrm{thankyou}\mathbf{S}\mathrm{er}$$

Commented by mnjuly1970 last updated on 23/Sep/21

$$thsnkzalotsir$$

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