Question

Prove that cos 24 . cos 48 . cos 96 . cos 168 = 1/16

Answer 1

$\cos 24\cdot \cos 48\cdot \cos 96\cdot \cos 192 = \frac{1}{16}$ Proved.

Step-by-step explanation:

Consider the provided information.

$\cos 24\cdot \cos 48\cdot \cos 96\cdot \cos 192 = \frac{1}{16}$

Consider the LHS.

$\cos 24\cdot \cos 48\cdot \cos 96\cdot \cos 192$

Divide and multiply 2 sin 24.

$\dfrac{2\sin 24}{2\sin 24}(\cos 24\cdot \cos 48\cdot \cos 96\cdot \cos 192)$

$\dfrac{1}{2\sin 24}(2\sin 24\cos 24\cdot \cos 48\cdot \cos 96\cdot \cos 192)$

Use the identity: sin 2θ=2sinθcosθ

$\dfrac{1}{2\sin 24}(\sin 48\cdot \cos 48\cdot \cos 96\cdot \cos 192)$

Multiply numerator and denominator by 2.

$\dfrac{1}{4\sin 24}(2\sin 48\cdot \cos 48\cdot \cos 96\cdot \cos 192)$

Use the identity: sin 2θ=2sinθcosθ

$\dfrac{1}{4\sin 24}(\sin 96\cdot \cos 96\cdot \cos 192)$

$\dfrac{1}{8\sin 24}(2\sin 96\cdot \cos 96\cdot \cos 192)$

$\dfrac{1}{8\sin 24}(\sin 192\cdot \cos 192)$

$\dfrac{1}{16\sin 24}(2\sin 192\cdot \cos 192)$

$\dfrac{1}{16\sin 24}(\sin 384)$

$\dfrac{1}{16\sin 24}(\sin 360+24)$

$\dfrac{1}{16\sin 24}(\sin 24)$

$\dfrac{1}{16}$

LHS=RHS

Hence, proved

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