((4sin (((2π)/7))+sec ((π/(14))))/(cot ((π/7)))) ?


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Question Number 125833 by bramlexs22 last updated on 14/Dec/20

$\frac{4 \sin (\frac{2 π}{7}) + \sec (\frac{π}{14})}{\cot (\frac{π}{7})} ?$
	Answered by Dwaipayan Shikari last updated on 14/Dec/20

	$\frac{4 s i n \frac{2 π}{7} c o s \frac{π}{14} s i n \frac{π}{7} + s i n \frac{π}{7}}{c o s \frac{π}{14} c o s \frac{π}{7}} = \frac{2 (c o s \frac{π}{7} - c o s \frac{3 π}{7}) c o s \frac{π}{14} + s i n \frac{π}{7}}{c o s \frac{π}{14} c o s \frac{π}{7}}$ $= \frac{c o s \frac{π}{14} - c o s \frac{3 π}{14} - c o s \frac{5 π}{14} + c o s \frac{π}{2} + c o s \frac{5 π}{14}}{c o s \frac{π}{14} c o s \frac{π}{7}} s i n \frac{π}{7} = c o s \frac{5 π}{14}$ $= \frac{c o s \frac{π}{14} - c o s \frac{3 π}{7}}{c o s \frac{π}{14} c o s \frac{π}{7}} = 2 (\frac{c o s \frac{π}{14} c o s \frac{π}{7}}{c o s \frac{π}{14} c o s \frac{π}{7}}) = 2$
	Answered by liberty last updated on 14/Dec/20

	$l e t \frac{π}{14} = x t h e n \frac{4 \sin 4 x + \sec x}{\cot 2 x} =$ $\frac{4 \sin 4 x \cos x + 1}{\cos x (\frac{\cos 2 x}{\sin 2 x})} = \frac{4 \sin 4 x \cos x \sin 2 x + \sin 2 x}{\cos x \cos 2 x}$ $= \frac{- 2 (\cos 6 x - \cos 2 x) \cos x + \sin 2 x}{\cos x \cos 2 x}$ $= \frac{2 \cos 2 x \cos x - 2 \cos 6 x \cos x + \sin 2 x}{\cos x \cos 2 x}$ $= \frac{2 \cos 2 x \cos x - \cos 7 x - \cos 5 x + \sin 2 x}{\cos x \cos 2 x}$ $[w e k n o w t h a t \cos 7 x = \cos \frac{π}{2} = 0 \land \cos 5 x = \cos \frac{5 π}{14} = \sin \frac{2 π}{14}]$ $t h u s w e g e t \cos 7 x - \cos 5 x + \sin 2 x = 0$ $s o \frac{2 \cos 2 x \cos x + 0}{\cos x \cos 2 x} = 2 .$
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