SATHEE: Trigonometrical Ratios and Identities 1 Question 26

Trigonometrical Ratios and Identities 1 Question 26

26. Show that $16 \cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{8 π}{15} \cos \frac{16 π}{15} = 1$

$(1983, 2 M)$

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Solution:

$16 \cos \frac{2 π}{15} \cdot \cos \frac{4 π}{15} \cdot \cos \frac{8 π}{15} \cdot \cos \frac{16 π}{15}$

$= 16 (\cos A \cdot \cos 2 A \cos 2^{2} A \cdot \cos 2^{3} A)$

where, $A = \frac{2 π}{15}$

$= 16 \frac{\sin 2^{4} A}{2^{4} \sin A} = \frac{\sin 2^{4} \frac{2 π}{15}}{\sin \frac{2 π}{15}}$

$= \frac{\sin \frac{32 π}{15}}{\sin \frac{2 π}{15}} = \frac{\sin 2 π + \frac{2 π}{15}}{\sin \frac{2 π}{15}}$

$= \frac{\sin \frac{2 π}{15}}{\sin \frac{2 π}{15}} = 1$

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Trigonometrical Ratios and Identities 1 Question 26

26. Show that 16cos⁡2π15cos⁡4π15cos⁡8π15cos⁡16π15=1

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26. Show that $16 \cos \frac{2 π}{15} \cos \frac{4 π}{15} \cos \frac{8 π}{15} \cos \frac{16 π}{15} = 1$