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Prove that, tan20° tan40° tan80° = $\sqrt{3}$

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Prove that, tan20° tan40° tan80° = $\sqrt{3}$

L.H.S:

$\tan 20°tan40°tan80°$
= $\frac{\sin 20°sin40°sin80°}{\cos 20°cos40°cos80°}$
= $\frac{2\sin 20°sin40°sin80°}{2\cos 20°cos40°cos80°}$
= $\frac{{ cos(20° – 40°) – cos(20° + 40°)} sin80°}{{ cos(20° + 40°) + cos(20° – 40°)} cos80°}$
= $\frac{(\cos 20° – cos60°)sin80°}{(cos60° + cos20°)cos80}$
= $\frac{(cos20° – \frac{1}{2})sin80°}{(\frac{1}{2} + cos20°)cos80°}$
= $\frac{\cos 20°sin80° – \frac{1}{2}\sin 80°}{\frac{1}{2}cos80° + cos20°cos80°}$
= $\frac{2sin80°cos20° – sin80}{\cos 80° + 2cos20°cos80°}$
= $\frac{\sin (80° + 20°) + sin(80° – 20°) – sin80°}{\cos 80° + cos(20° + 80°) + cos(20° – 80°)}$
= $\frac{\sin 100°+ sin60° – sin(90° – 10°)}{\cos (90° – 10°) + cos100° + cos60°}$
= $\frac{\sin 100° + sin60° – cos10°}{\sin 10° + cos100° + cos60°}$
= $\frac{\sin (90° + 10°) + sin60° – cos10°}{\sin 10° + cos(90° + 10°) + cos60°}$
= $\frac{\cos 10° + sin60° – cos10°}{\sin 10° – sin10° + cos60°}$
= $\frac{\sin 60°}{\cos 60°}$
= tan60°
= $\sqrt{3}$ =R.H.S [Proved]

আরও দেখুন

  1. Prove that cos 80° – cos40° + $\sqrt{3}$ cos70° = 0
  2. Prove that, 8 sin20° sin40° sin80° = $\sqrt{3}$
  3. If sinθ + sinΦ=a and cos θ +cosΦ=b prove that $tan\frac{\theta  – \phi }{2}$ =  $\pm \sqrt{\frac{4 – a^2 – b^2}{a^2 + b^2}}$
  4. Prove that,4 sin15° sin75°=$\sqrt{2}$ (cos 105° + sin75°)
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