Cos 78-cos 42-cos 36-1-8

L.H.S = cos 78^∘ cos 42^∘ cos 36^∘ = (2 cos 78^∘. cos 42^∘)/2. cos 36^∘ = 1/2 (cos 120^∘ + cos 36^∘). cos 36^∘ =1/2 ( 1/2+√(5)+1/4 ) √(5)+1/4 =1/8 [ 2 (√(5) ...

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Byju's Answer

Standard IX

Mathematics

Values of Trigonometric Ratios

cos 78∘cos 42...

Question

$c o s 78^{\circ} c o s 42^{\circ} c o s 36^{\circ} = \frac{1}{8}$

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Solution

$L . H . S = c o s 78^{\circ} c o s 42^{\circ} c o s 36^{\circ} = \frac{(2 c o s 78^{\circ} . c o s 42^{\circ})}{2} . c o s 36^{\circ} = \frac{1}{2} (c o s 120^{\circ} + c o s 36^{\circ}) . c o s 36^{\circ} = \frac{1}{2} (\frac{- 1}{2} + \frac{\sqrt{5} + 1}{4}) \frac{\sqrt{5} + 1}{4} = \frac{1}{8} \frac{[- 2 (\sqrt{5} + 1) + 5 + 1 + 2 \sqrt{5}]}{4} = \frac{1}{8} [\frac{4}{4}] = \frac{1}{8}$
=RHS

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cos 78∘ cos 42∘ cos 36∘=18

L.H.S=cos 78∘ cos 42∘ cos 36∘=(2 cos 78∘. cos 42∘)2. cos 36∘=12 (cos 120∘+cos 36∘). cos 36∘=12 (−12+√5+14)√5+14=18 [−2(√5+1)+5+1+2√5]4=18 [44]=18 =RHS

$c o s 78^{\circ} c o s 42^{\circ} c o s 36^{\circ} = \frac{1}{8}$