Question

17) Prove That Cos 20 Cos 40 Cos 60 Сos 80 = 1/16

Answer 1

Step-by-step explanation:

Formula used

$2\sin( \theta) \cos( \theta) = \sin(2 \theta)$

Now you question

$\:\: \cos(20). \cos(40) . \cos(60) . \cos(80) \\ \\ = \cos(20). \cos(40) . \frac{1}{2} . \cos(80) \\ \\ = \frac{1}{2 \times 2 \sin(20) } (2 \sin(20) \cos(20) ). \cos(40) . \cos(80) \\ \\ = \frac{1}{4 \sin(20) } ( \sin(40) \cos(40) ). \cos(80) \\ \\ = \frac{1}{4 \sin(20) } \times \frac{1}{2} (2 \sin(40) \cos(40) ). \cos(80) \\ \\ = \frac{1}{8 \sin(20) } ( \sin(80) \cos(80)) \\ \\ = \frac{1}{16 \sin(20) } \sin(160) \\ \\ = \frac{1}{16 \sin(20) } \sin(180 - 20) \\ = \frac{1}{16 \cancel { \sin(20) }} \cancel { \sin(20) } \\ \\ = \frac{1}{16}$