cos80 + cos 70 + cos50 + cos40 = cos20 + cos10

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Solution

$L . H . S . = \cos 80 + \cos 70 + \cos 50 + \cos 40 = (\cos 80 + \cos 40) + (\cos 70 + \cos 50) = (2 \cos (\frac{80 + 40}{2}) \cos (\frac{80 - 40}{2})) + (2 \cos (\frac{50 + 70}{2}) \cos (\frac{50 - 70}{2})) [Using formula, cosC + cosD = 2 \cos (\frac{C + D}{2}) \cos (\frac{C - D}{2})] = (2 \cos 60 \cos (- 20)) + (2 \cos 60 \cos (- 10)) = (2 \times \frac{1}{2} \times (\cos 20)) + (2 \times \frac{1}{2} \times (\cos 10)) [We know that, \cos (- θ) = cosθ] = \cos 20 + \cos 10 = R . H . S . Hence Proved .$

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