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Simplify cos(a)cos(2a)cos(3a)...cos(999a) if a=(2pi)/1999

Simplify cos(a)cos(2a)cos(3a)...cos(999a) if a=(2pi)/1999

Grade:11

2 Answers

Nishant Vora IIT Patna
askIITians Faculty 2467 Points
5 years ago
multiply divide by 2 sina in numerator nd denomenator
so you will get


\frac{1}{2^{999}} \frac{sin (2*999)a}{sin a}

\frac{1}{2^{999}} \frac{sin (1998)a}{sin a}
\frac{1}{2^{999}} \frac{sin (1999a - a)}{sin a}
\frac{1}{2^{999}} \frac{sin (2 pi - a)}{sin a}
-\frac{1}{2^{999}}
Piyush
15 Points
3 years ago
Let P=cosa cos2a cos3a.......cos999a 
Let Q=sina sin2a sin3a............sin999a
Now,
PQ×2^999=(2sinacosa)(2sin2acos2a)….......(2sin999acos999a)
=sin2a sin4a sin6a........sin1998a
=(sin2a sin4a....sin1998a){-sin(2π-100a)}{-sin(2π-1002a)}......{-sin(2π-1988a)}
=sin2a sin4a....sin998a sin999asin997a....sinx 
=Q
Hence P=1/2^999
 

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