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cos 20 + cos 40 + cos 60 + cos 80 = ?

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Correct Answer - Option 4 : None of these.

Concept:

Trigonometry:

Trigonometric Formula:

sin (A ± B)

sin A cos B ± sin B cos A

cos (A ± B)

cos A cos B ∓ sin A sin B

tan (A ± B)

\(\rm \dfrac{\tan A \pm \tan B}{1 \mp \tan A \tan B}\)

cot (A ± B)

\(\rm \dfrac{\mp1 +\cot A \cot B}{\pm\cot A+\cot B}\)


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 30 = \(\frac{\sqrt3}2\), sin 30 = \(\frac12\).

cos 20 + cos 40 + cos 60 + cos 80

\(\frac12\) + 2 \(\rm\left[\cos\left(\frac{40^\circ+20^\circ}{2}\right)\cos\left(\frac{40^\circ-20^\circ}{2}\right)\right]\) + cos (90 - 10)

\(\frac12\) + 2 cos 30 cos 10 + sin 10

\(\frac12\) + 2 cos 30 cos 10 + 2 sin 30 sin 10

\(\frac12\) + 2cos (30 - 10)

\(\frac12\) + 2cos 20

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