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cos 20 cos 40 cos 60 cos 80 = ?

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Correct Answer - Option 4 : None of these.

Concept:

Sum-Product formulae:

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

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

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

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

Calculation:

Recall that cos 60\(\frac12\).

cos 20 cos 40 cos 60 cos 80

\(\frac12\) (cos 20 cos 80) cos 40

= ​\(\frac14\) [cos (80 - 20) + cos (80 + 20)] cos 40

= \(\frac14\) [cos 60 + cos 100] cos 40

= \(\frac14\) [\(\frac12\) + cos 100] cos 40

\(\frac18\) cos 40 + \(\frac14\) [cos 100 cos 40]

\(\frac18\) cos 40 + \(\frac18\) [cos (100 + 40) + cos (100 - 40)]

\(\frac18\) cos 40 + \(\frac18\) [cos 140 + cos 60]

\(\frac18\) cos 40 + \(\frac18\) cos (180 - 40) + \(\frac18\) cos 60

\(\frac18\) cos 40 - \(\frac18\) cos 40 + \(\frac18\) cos 60

\(\frac1{16}\)

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