Question

Please someone help me....​

Answer 1

We are given the equation cos(20°)(cos(40°)(cos(60°)(cos(80°) = √3 / 8. Let's once again start by applying the identity 'sin(s)sin(t) = - cos(s + t) + cos(s - t) / 2. In this case if we focus on the expression 'cos(20°)(cos(40°),' s would be = 20°, and t = 40°.

`\mathrm{Use\:the\:following\:identity}:\quad \sin \left(s\right)\sin \left(t\right)=(-\cos \left(s+t\right)+\cos \left(s-t\right))/(2)`

`\sin \left(20^(\circ \:)\right)\sin \left(40^(\circ \:)\right)=(-\cos \left(20^(\circ \:)+40^(\circ \:)\right)+\cos \left(20^(\circ \:)-40^(\circ \:)\right))/(2)`

`\mathrm{Substitute}:(-\cos \left(20^(\circ \:)+40^(\circ \:)\right)+\cos \left(20^(\circ \:)-40^(\circ \:)\right))/(2)\sin \left(80^(\circ \:)\right)`

`\mathrm{Multiply\:fractions}:(\sin \left(80^(\circ \:)\right)\left(-\cos \left(60^(\circ \:)\right)+\cos \left(-20^(\circ \:)\right)\right))/(2)`

Remember that cos(- x) = cos(x). Respectively cos(- 20°) = cos(20°). Let's substitute and afterwards apply the identity 'cos(60°) = 1 / 2.'

`(\sin \left(80^(\circ \:)\right)\left(-\cos \left(60^(\circ \:)\right)+\cos \left(20^(\circ \:)\right)\right))/(2) = (\sin \left(80^(\circ \:)\right)\left(-(1)/(2)+\cos \left(20^(\circ \:)\right)\right))/(2)`

And if we further simplify the expression, we should receive the following...

`(\sin \left(80^(\circ \:)\right)\left(-1+2\cos \left(20^(\circ \:)\right)\right))/(4)`

Now we want to prove that this expression = √3 / 8. The denominator here is 4 so we can multiply the whole thing by 2 to have a denominator of 8. 2((sin(80°)(- 1 + 2cos(20°)) when simplified = √3. Therefore the expression is true.

Answer 2

Answer: see proof below

Explanation:

Use the following Product to Sum Identities:

2 sin A sin B = cos (A - B) - cos (A + B)

2 sin A cos B = sin (A + B) + sin (A - B)

Use the Unit Circle to evaluate: cos 120 = -1/2 & sin 60 = √3/2

Proof LHS → RHS

LHS: sin 20 · sin 40 · sin 80

Regroup: (1/2) sin 20 · 2 sin 40 · sin 80

Product to Sum Identity: (1/2) sin 20 [cos(80-40) - cos (80+40)]

Simplify: (1/2) sin 20 [cos 40 - cos 120]

Unit Circle: (1/2) sin 20 [cos 40 + (1/2)]

Distribute: (1/2) sin 20 cos 40 + (1/4) sin 20

Product to Sum Identity: (1/4)[sin(20 + 40) + sin (20 - 40)] + (1/4) sin 20

Simplify: (1/4)[sin 60 + sin (-20)] + (1/4) sin 20

= (1/4)[sin 60 - sin 20] + (1/4) sin 20

Unit Circle: (1/4)[(√3/2) - sin 20] + (1/4) sin 20

Distribute: (√3/8) - (1/4) sin 20 + (1/4) sin 20

Simplify: √3/8

LHS = RHS: √3/8 = √3/8
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