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Question: \(( a + b + c ) ( \cos A + \cos B + \cos C ) =\)...

(a+b+c)(cos⁡A+cos⁡B+cos⁡C)=( a + b + c ) ( \cos A + \cos B + \cos C ) =

A

∑asin⁡2A2\sum a \sin ^ { 2 } \frac { A } { 2 }

B

∑acos⁡2A2\sum a \cos ^ { 2 } \frac { A } { 2 }

C

2∑asin⁡2A22 \sum a \sin ^ { 2 } \frac { A } { 2 }

D

2∑acos⁡2A22 \sum a \cos ^ { 2 } \frac { A } { 2 }

Answer

2∑acos⁡2A22 \sum a \cos ^ { 2 } \frac { A } { 2 }

Explanation

Solution

(a+b+c)(cos⁡A+cos⁡B+cos⁡C)=9( a + b + c ) ( \cos A + \cos B + \cos C ) = 9 terms.

=

=acos⁡A+bcos⁡B+ccos⁡C+(a+b+c)a \cos A + b \cos B + c \cos C + ( a + b + c )

= ∑a(1+cos⁡A)=2∑acos⁡2A2\sum a ( 1 + \cos A ) = 2 \sum a \cos ^ { 2 } \frac { A } { 2 }.