(sin20°sin50°sin70°)/(sin80°)

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The expression can be simplified as follows:

sin(20°)sin(50°)sin(70°) / sin(80°)

Using the trigonometric identity:

sin(α)sin(β) = 1/2[cos(α-β) - cos(α+β)]

Letting α = 70° and β = 50°:

sin(70°)sin(50°) = 1/2[cos(70° - 50°) - cos(70° + 50°)] = 1/2[cos(20°) - cos(120°)] = 1/2[cos(20°) + 1/2]

Now we have:

(sin(20°)(1/2[cos(20°) + 1/2])) / sin(80°)

Using the double angle identity:

cos(2θ) = 2cos^2(θ) - 1

Let θ = 20°:

cos(40°) = 2cos^2(20°) - 1

Rearranging:

cos^2(20°) = (1 + cos(40°)) / 2

Substitute back to the expression:

(sin(20°)(1/2[cos(20°) + 1/2])) / sin(80°)
= (sin(20°)(1/2[cos(20°) + 1/2])) / (2sin(40°)cos(40°))

= (1/2[sin(20°)cos(20°) + sin(20°)/2]) / (2sin(40°)cos(40°))

= (1/2[1/2(sin(40°)) + sin(20°)/2]) / (2sin(40°)cos(40°))

= [1/(4cos(40°)) + sin(20°)/(4cos(40°))] / (2sin(40°)cos(40°))

= [1/(4sin(50°)) + sin(20°)/(4sin(50°))] / (2sin(50°))

= [1 + 2sin(20°)] / (8sin(50°))

Now, you can calculate the final value using a calculator.