Question

cos(1)+cos(2)+...........+cos(89)=?



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Answer 1

Answer:

we know that

2sin x cos x= sin 2x

To convert the expression,we have to multiply and divide 2 ,44 times

\begin{gathered} \frac{2 \sin(1°) \cos(1°) \times2 \sin(2°) \cos(2°) \times 2 \sin(3°) \cos(3°) \times...2 \sin(44°) \cos(44°) \times \: cos45° }{ {2}^{44} } \\ \\ = \frac{ \sin(2°) \times \sin(4°) \times \sin(6°)... \sin(88°) \times \frac{1}{ \sqrt{2} } }{ {2}^{44} } \\ \\ \cos(1°) \cos(2°) ... \cos(88°) \cos(89°) \\\\= \frac{1}{ \sqrt{2} \times {2}^{44} } \sin(2°) \times \sin(4°) \times \sin(6°)... \sin(88°) \\ \end{gathered}

2

44

2sin(1°)cos(1°)×2sin(2°)cos(2°)×2sin(3°)cos(3°)×...2sin(44°)cos(44°)×cos45°

=

2

44

sin(2°)×sin(4°)×sin(6°)...sin(88°)×

2

1

cos(1°)cos(2°)...cos(88°)cos(89°)

=

2

×2

44

1

sin(2°)×sin(4°)×sin(6°)...sin(88°)

Hope it helps you