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3.Trigonometrical Ratios, Functions and Identities
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The value of $cos\, \frac{\pi }{{10}} \,cos\, \frac{2\pi }{{10}} \,cos\,\frac{4\pi }{{10}}\, cos\,\frac{8\pi }{{10}}\, cos\,\frac{16\pi }{{10}}$ is
A
$\frac{1 }{{32}}$
B
$\frac{1}{{16}}$
C
$\frac{{\cos \,\,\left( {\pi /10} \right)}}{{16}}$
D
$-\frac{{\sqrt {10\,\, + \,\,2\sqrt 5 } }}{{64}}$
Solution
$\pi /10 = \theta$
$E =$$\frac{{2\,\sin \theta \,(\cos \theta \,.\,\cos 2\theta \,.\,\cos 4\theta \,.\cos 8\theta \,.\cos 16\theta \,)}}{{2\,\sin \theta }}$
$=$$\frac{{\sin 32\theta }}{{32\,\sin \theta }}$
$=$$\frac{{\sin (30\theta \, + \,2\theta )}}{{32.\,\sin \theta }}$ $=$ $ – \,\,\frac{1}{{16}}\,\cos \frac{\pi }{{10}}$
$=$ $ – \,\,\frac{1}{{16}}\,\,\sqrt {1 – {{\sin }^2}\pi /10} $
Standard 11
Mathematics
