1-cos-81-cos3 -81-cos5 -81-cos7 -8-

The correct option is C (1+cosπ/8)(1+cos3π/8)(1+cos5π/8)(1+cos7π/8) =(1+cosπ/8+cos7π/8+cosπ/8cos7π/8) (1+cos5π/8+cos3π/8+cos3π/8cos5π/8) =(1+cosπ/8 cosπ/8+cosπ ...

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Byju's Answer

1+cosπ/81+cos...

Question

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

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Solution

The correct option is C
$(1 + c o s \frac{π}{8}) (1 + c o s \frac{3 π}{8}) (1 + c o s \frac{5 π}{8}) (1 + c o s \frac{7 π}{8})$
$= (1 + c o s \frac{π}{8} + c o s \frac{7 π}{8} + c o s \frac{π}{8} c o s \frac{7 π}{8})$
$(1 + c o s \frac{5 π}{8} + c o s \frac{3 π}{8} + c o s \frac{3 π}{8} c o s \frac{5 π}{8})$
$= (1 + c o s \frac{π}{8} - c o s \frac{π}{8} + c o s \frac{π}{8} c o s \frac{7 π}{8})$
$(1 + c o s \frac{5 π}{8} - c o s \frac{5 π}{8} + c o s \frac{3 π}{8} c o s \frac{5 π}{8})$
$= (1 + c o s \frac{π}{8} c o s \frac{7 π}{8}) (1 + c o s \frac{3 π}{8} c o s \frac{5 π}{8})$
$= \frac{1}{4} (2 + 2 c o s \frac{π}{8} c o s \frac{7 π}{8}) (2 + 2 c o s \frac{3 π}{8} c o s \frac{5 π}{8})$
$= \frac{1}{4} (2 + c o s \frac{3 π}{4} + c o s π) (2 + c o s \frac{π}{4} + c o s π)$
$= \frac{1}{4} (1 + c o s \frac{3 π}{4}) (1 + c o s \frac{π}{4}) = \frac{1}{4} (1 - c o s \frac{π}{4}) (1 + c o s \frac{π}{4})$
$= \frac{1}{4} (1 - c o s^{2} \frac{π}{4}) = \frac{1}{4} (1 - \frac{1}{2}) = \frac{1}{8} .$
Aliter: $(1 + c o s \frac{π}{8}) (1 + c o s \frac{7 π}{8}) (1 + c o s \frac{3 π}{8}) (1 + c o s \frac{5 π}{8})$
$= (1 + c o s \frac{π}{8}) (1 - c o s \frac{π}{8}) (1 + c o s \frac{3 π}{8}) (1 - c o s \frac{3 π}{8})$
$= (1 - c o s^{2} \frac{π}{8}) (1 - c o s^{2} \frac{3 π}{8}) = s i n^{2} \frac{π}{8} s i n^{2} \frac{3 π}{8}$
$= \frac{1}{4} {(2 s i n \frac{π}{8} . s i n \frac{3 π}{8})}^{2} = \frac{1}{4} {(c o s \frac{π}{4} - c o s \frac{π}{2})}^{2} = \frac{1}{8} .$

Suggest Corrections

(1+cosπ8)(1+cos3π8)(1+cos5π8)(1+cos7π8)=

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