Tan20 tan40 tan60 tan80= 3. Prove it
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tan20°tan40°tan60°tan80°
={(sin20°sin40°sin80°)/(cos20°cos40°cos80°)}×√3
=√3×{(2sin20°sin40°sin80°)/(2cos20°cos40°cos80°)}
=√3×[{cos(20°-40°)-cos(20°+40°)}sin80°/{cos(20°-40°)+cos(20°+40°)}cos80°]
=√3×[{cos20°sin80°-(1/2)sin80°}/{cos20°cos80°+(1/2)cos80°}] [∵, cos60°=1/2]
=√3×[{sin100°-sin(-60°)-sin80°}/{cos60°+cos100°+cos80°}]
=√3×[{sin100°-sin80°+√3/2}/{1/2+cos100°+cos80°}]
=√3×[{2cos(100°+80°)/2sin(100°-80°)/2+√3/2}/{1/2+2cos(100°+80°)/2cos(100°-80°)/2}]
=√3×√3/2×2/1
=3
Hence Proved.
={(sin20°sin40°sin80°)/(cos20°cos40°cos80°)}×√3
=√3×{(2sin20°sin40°sin80°)/(2cos20°cos40°cos80°)}
=√3×[{cos(20°-40°)-cos(20°+40°)}sin80°/{cos(20°-40°)+cos(20°+40°)}cos80°]
=√3×[{cos20°sin80°-(1/2)sin80°}/{cos20°cos80°+(1/2)cos80°}] [∵, cos60°=1/2]
=√3×[{sin100°-sin(-60°)-sin80°}/{cos60°+cos100°+cos80°}]
=√3×[{sin100°-sin80°+√3/2}/{1/2+cos100°+cos80°}]
=√3×[{2cos(100°+80°)/2sin(100°-80°)/2+√3/2}/{1/2+2cos(100°+80°)/2cos(100°-80°)/2}]
=√3×√3/2×2/1
=3
Hence Proved.
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