Question

Please someone help me to prove this...
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Answer 1

In this case we have the equation cos(20°)(cos(40°)(cos(60°)(cos(80°) = 1 / 16, and are asked to prove that this equation is true. Let's start by using the property 'cos(s)cos(t) = cos(s + t) + cos(s - t) / 2.' Taking the bit 'cos(20°)(cos(40°)' we know that s should = 20°, and t should = 40°.

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

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

`\mathrm{Substituting\:the\:value\:back}:(\cos \left(20^(\circ \:)+40^(\circ \:)\right)+\cos \left(20^(\circ \:)-40^(\circ \:)\right))/(2)\cos \left(60^(\circ \:)\right)\cos \left(80^(\circ \:)\right)`

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

This expression is indeed not simplified, but remember that we can use the identities 'cos(- 20°) = cos(20°)' and 'cos(60°) = 1 / 2.' Let's substitute one step and a time.

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

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

`((1)/(2)\cos \left(80^(\circ \:)\right)\left((1)/(2)+\cos \left(20^(\circ \:)\right)\right))/(2) = (\cos \left(80^(\circ \:)\right)\left(1+2\cos \left(20^(\circ \:)\right)\right))/(8)`

Now that we have this 'simplified expression,' if we take the numerator as a whole in decimal form, it will = 0.5. Respectively 0.5 / 8 = 1 / 2 / 8 = 1 / 16. Hence our equation is true.

`\cos \left(80^(\circ \:\:)\right)\left(1+2\cos \left(20^(\circ \:\:)\right)\right) = 0.5,\\0.5 / 8 = (1 / 2) / 8 = 1 / 16`

Answer 2

Explanation:

cos 20° cos 40° cos 60° cos 80°

Simplify cos 60° to ½.

½ cos 20° cos 40° cos 80°

We can use double angle formula, sin(2θ) = 2 sin θ cos θ. Multiply and divide by 2 sin 20°.

½ (2 sin 20° cos 20° cos 40° cos 80°) / (2 sin 20°)

½ (sin 40° cos 40° cos 80°) / (2 sin 20°)

Multiply and divide by 2 and use double angle formula again.

½ (2 sin 40° cos 40° cos 80°) / (4 sin 20°)

½ (sin 80° cos 80°) / (4 sin 20°)

Multiply and divide by 2 and use double angle formula again.

½ (2 sin 80° cos 80°) / (8 sin 20°)

½ (sin 160°) / (8 sin 20°)

Use phase shift identity sin θ = sin(180−θ).

½ (sin 20°) / (8 sin 20°)

1/16