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Trigonometry> Simplify cos(a)cos(2a)cos(3a)...cos(999a)...

question mark

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

gdhh , 8 Years ago

Grade 11

2 Answers

Nishant Vora

Last Activity: 8 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

Last Activity: 6 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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