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Question Number 114577 by Khalmohmmad last updated on 19/Sep/20

$$\mathrm{x}=\frac{\pi }{30}$$$$\frac{\cos 14\mathrm{x}\times \cos 6\mathrm{x}}{\cos 4\mathrm{x}}=?$$

Commented by bemath last updated on 19/Sep/20

$$\frac{\cos 20x+\cos 8x}{2\cos 4x}=\frac{\cos \frac{2\pi }{3}+\cos \frac{4\pi }{15}}{2\cos \frac{2\pi }{15}}$$$$=\frac{-\frac{1}{2}+2\cos {}^2\left(\frac{2\pi }{15}\right)-1}{2\cos \frac{2\pi }{15}}$$

Commented by MJS_new last updated on 19/Sep/20

$$\mathrm{we}\mathrm{can}\mathrm{express}\mathrm{this}\mathrm{with}\mathrm{roots}\mathrm{but}\mathrm{I}{\mathrm{haven}}^{\prime }\mathrm{t}$$$$\mathrm{got}\mathrm{the}\mathrm{time}\mathrm{right}\mathrm{now}$$

Commented by MJS_new last updated on 20/Sep/20

$$\mathrm{I}\mathrm{get}\frac{13+7\sqrt{5}-\sqrt{390+174\sqrt{5}}}{8}$$

Commented by bobhans last updated on 20/Sep/20

$$howsir?$$

Commented by MJS_new last updated on 20/Sep/20

$$\frac{\pi }{30}=6°$$$$\cos 84°=\sin 6°=\sin (36°-30°)$$$$\cos 36°=\frac{1+\sqrt{5}}{4}$$$$\cos 24°=\cos (60°-36°)$$$$\left(\mathrm{or}\mathrm{use}\mathrm{other}\mathrm{sums}/\mathrm{differences}\right)$$$$\mathrm{now}\mathrm{apply}\mathrm{formulas}\mathrm{for}\sin (a\pm b),\cos (a\pm b)$$

Answered by 1549442205PVT last updated on 20/Sep/20

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\cos 14\mathrm{x}\times \cos 6\mathrm{x}}{\cos 4\mathrm{x}}=\frac{\cos 20\mathrm{x}+\cos 8\mathrm{x}}{2\cos 4\mathrm{x}}$$$$\mathrm{f}\left(\frac{\pi }{30}\right)=\frac{\cos 20\mathrm{x}+{2\cos }^{2}4\mathrm{x}-1}{2\cos 4\mathrm{x}}$$$$=\frac{\cos \frac{2\pi }{3}+{2\cos }^2\frac{2\pi }{15}-1}{2\cos \frac{2\pi }{15}}=\frac{-\frac{1}{2}+{2\cos }^{2}24°-1}{2\cos 24°}$$

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