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Integration by Substitution

Prove that ...

Question

Prove that $sin \frac{π}{15} + sin \frac{4 π}{15} - sin \frac{11 π}{15} - sin \frac{14 π}{15} = 0$

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Solution

$L . H . S . = sin \frac{π}{15} + sin \frac{4 π}{15} - sin \frac{11 π}{15} - sin \frac{14 π}{15} = sin 12^{o} + sin 48^{o} - sin 132^{o} - sin 168^{o} = sin 12^{o} + sin 48^{o} - sin (180^{o} - 48^{o}) - sin (180^{o} - 12^{o}) = sin 12^{o} + sin 48^{o} - sin 48^{o} - sin 12^{o} = sin 12^{o} - sin 12^{o} + sin 48^{o} - sin 48^{o} = 0 + 0 = 0 = R . H . S .$

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sin \frac{π}{15} + sin \frac{4 π}{15} - sin \frac{11 π}{15} - sin \frac{14 π}{15} = m

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sin \frac{π}{15} + sin \frac{4 π}{15} - sin \frac{11 π}{15} - sin \frac{14}{15} = 0

Q. Prove that :
4cos2A
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\cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{7 π}{15} = \frac{1}{16}

\cos \frac{π}{15} \cos \frac{2 π}{15} \cos \frac{3 π}{15} \cos \frac{4 π}{15} \cos \frac{5 π}{15} \cos \frac{6 π}{15} \cos \frac{7 π}{15} = \frac{1}{128}

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