Prove-that-sec-2-7-sec-4-7-sec-8-7-4-

Question Number 165558 by som(math1967) last updated on 03/Feb/22

P r o v e t h a t

s e c \frac{2 π}{7} + s e c \frac{4 π}{7} + s e c \frac{8 π}{7} = - 4

Answered by Rohit143Jo last updated on 04/Feb/22

A n s :

L e t, x = \frac{2 π}{7} \Leftrightarrow 7 x = 2 π

L H S, S e c (x) + S e c (2 x) + S e c (4 x)

\Rightarrow \frac{1}{C o s (x)} + \frac{1}{C o s (2 x)} + \frac{1}{C o s (4 x)}

\Rightarrow \frac{C o s (2 x) . C o s (4 x) + C o s (x) . C o s (4 x) + C o s (x) . C o s (2 x)}{C o s (x) . C o s (2 x) . C o s (4 x)}

\Rightarrow \frac{2 . C o s (2 x) . C o s (4 x) + 2 . C o s (x) . C o s (4 x) + 2 . C o s (x) . C o s (2 x)}{2 . C o s (x) . C o s (2 x) . C o s (4 x)}

\Rightarrow \frac{S i n (x) [C o s (6 x) + C o s (2 x) + C o s (5 x) + C o s (3 x) + C o s (3 x) + C o s (x)]}{2 S i n (x) C o s (x) . C o s (2 x) C o s (4 x)}

\Rightarrow \frac{2 S i n (x) [C o s (2 π - x) + C o s (2 x) + C o s (2 π - 2 x) + C o s (2 π - 4 x) + C o s (2 π - 4 x) + C o s (x)]}{2 S i n (2 x) C o s (2 x) . C o s (4 x)}

\Rightarrow \frac{4 S i n (x) [C o s (x) + C o s (2 x) + C o s (2 x) + C o s (4 x) + C o s (4 x) + C o s (x)]}{2 S i n (4 x) C o s (4 x)}

\Rightarrow \frac{4 [2 S i n (x) C o s (x) + 2 C o s (2 x) S i n (x) + 2 C o s (4 x) S i n (x)]}{S i n (8 x)}

\Rightarrow \frac{4 [S i n (2 x) + S i n (3 x) - S i n (x) + S i n (5 x) - S i n (3 x)]}{S i n (8 x)}

\Rightarrow \frac{4 [- S i n (x) + S i n (2 x) + S i n (2 π - 2 x)]}{S i n (2 π + x)}

\Rightarrow \frac{4 [- S i n (x) + S i n (2 x) - S i n (2 x)]}{S i n (x)}

\Rightarrow \frac{4 [- S i n (x)]}{S i n (x)}

\Rightarrow - 4, L H S (P r o v e d)

Commented by som(math1967) last updated on 04/Feb/22

V e r y n i c e s o l u t i o n t h a n k y o u

Commented by Rohit143Jo last updated on 04/Feb/22

W e l c o m e \dots \dots

Commented by peter frank last updated on 06/Feb/22

Thank you