Question Number 161254 by cortano last updated on 15/Dec/21
$$\:\frac{\mathrm{4sin}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)+\mathrm{sec}\:\left(\frac{\pi}{\mathrm{14}}\right)}{\mathrm{cot}\:\left(\frac{\pi}{\mathrm{7}}\right)}=? \\ $$
Commented by bobhans last updated on 15/Dec/21
![P = ((4sin (((2π)/7)) cos ((π/(14)))+1)/(cos ((π/(14)))[((cos ((π/7)))/(2sin ((π/(14))) cos ((π/(14)))))])) P = ((4sin (((2π)/7)) sin ((π/7))+2sin ((π/(14))))/(cos ((π/7)))) P = ((−2(cos (((3π)/7))−cos ((π/7)))+2cos (((3π)/7)))/(cos ((π/7)))) P = ((2cos ((π/7)))/(cos ((π/7)))) = 2](https://www.tinkutara.com/question/Q161288.png)
$$\mathcal{P}\:=\:\frac{\mathrm{4sin}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{14}}\right)+\mathrm{1}}{\mathrm{cos}\:\left(\frac{\pi}{\mathrm{14}}\right)\left[\frac{\mathrm{cos}\:\left(\frac{\pi}{\mathrm{7}}\right)}{\mathrm{2sin}\:\left(\frac{\pi}{\mathrm{14}}\right)\:\mathrm{cos}\:\left(\frac{\pi}{\mathrm{14}}\right)}\right]} \\ $$$$\:\mathcal{P}\:=\:\frac{\mathrm{4sin}\:\left(\frac{\mathrm{2}\pi}{\mathrm{7}}\right)\:\mathrm{sin}\:\left(\frac{\pi}{\mathrm{7}}\right)+\mathrm{2sin}\:\left(\frac{\pi}{\mathrm{14}}\right)}{\mathrm{cos}\:\left(\frac{\pi}{\mathrm{7}}\right)} \\ $$$$\:\mathcal{P}\:=\:\frac{−\mathrm{2}\left(\mathrm{cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)−\mathrm{cos}\:\left(\frac{\pi}{\mathrm{7}}\right)\right)+\mathrm{2cos}\:\left(\frac{\mathrm{3}\pi}{\mathrm{7}}\right)}{\mathrm{cos}\:\left(\frac{\pi}{\mathrm{7}}\right)} \\ $$$$\:\mathcal{P}\:=\:\frac{\mathrm{2cos}\:\left(\frac{\pi}{\mathrm{7}}\right)}{\mathrm{cos}\:\left(\frac{\pi}{\mathrm{7}}\right)}\:=\:\mathrm{2}\: \\ $$