prove-that-sin-16x-cot-x-1-2cos-2x-2cos-4x-2cos-6x-2cos-16x-

Question Number 97135 by M±th+et+s last updated on 06/Jun/20

p r o v e t h a t :

s i n (16 x) c o t (x) = 1 + 2 c o s (2 x) + 2 c o s (4 x) + 2 c o s (6 x) + \dots + 2 c o s (16 x)

Commented by prakash jain last updated on 07/Jun/20

\cos x = \frac{1}{2} (e^{i x} + e^{- i x})

RHS = 1 + \underset{k = 1}{\sum^{8}} (e^{2 k i x} + e^{- 2 k i x})

= e^{- 16 i x} + e^{- 14 k x} + \dots + e^{16 i x}

= \frac{e^{- 16 i x} (e^{34 i x} - 1)}{e^{2 i x} - 1} = \frac{e^{18 i x} - e^{- 16 i x}}{e^{2 i x} - 1}

\sin 16 x \cot x = \frac{e^{16 i x} - e^{- 16 i x}}{2} \times \frac{e^{i x} + e^{- i x}}{e^{i x} - e^{- i x}}

= \frac{e^{16 x} - e^{- 16 x}}{2} \times \frac{e^{2 i x} + 1}{e^{2 i x} - 1}

= \frac{e^{18 i x} - e^{- 14 i x} + e^{16 i x} - e^{- 16 i x}}{2 (e^{2 i x} - 1)}

Equality does not seem be valid .

I will recheck my calculation again .