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Complementary Trigonometric Ratios

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Question

Show that :

$cos \frac{π}{8} + cos \frac{3 π}{8} + cos \frac{5 π}{8} + cos \frac{7 π}{8} = 0$ .

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Solution

$cos \frac{π}{8} + cos \frac{3 π}{8} + cos \frac{5 π}{8} + cos \frac{7 π}{8} = 0$

L.H.S:-
$cos (\frac{π}{8}) + cos (\frac{3 π}{8}) + cos (π - \frac{3 π}{8}) + cos (π - \frac{π}{8})$

$= 0$

$=$ R.H.S

LHS $=$ RHS

$∴ cos \frac{π}{8} + cos \frac{3 π}{8} + cos \frac{5 π}{8} + cos \frac{7 π}{8} = 0$ .

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(1 + cos \frac{π}{8}) (1 + cos \frac{3 π}{8}) (1 + cos \frac{5 π}{8}) (1 + cos \frac{7 π}{8}) = \frac{1}{8}

(1 + cos \frac{π}{8}) (1 + cos \frac{3 π}{8}) (1 + cos \frac{5 π}{8}) (1 + cos \frac{7 π}{8}) = \frac{1}{8} .

(1 + cos \frac{π}{8}) (1 + cos \frac{3 π}{8}) (1 + cos \frac{5 π}{8}) (1 + cos \frac{7 π}{8}) = \frac{1}{8}

(1 + cos \frac{π}{8}) (1 + cos \frac{3 π}{8}) (1 + cos \frac{5 π}{8}) (1 + cos \frac{7 π}{8}) = \frac{1}{8}

(1 + c o s \frac{π}{8}) (1 + c o s \frac{3 π}{8}) (1 + c o s \frac{5 π}{8}) (1 + c o s \frac{7 π}{8}) =

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