# Cos \pi /11 +cos3 \pi /11 +cos5 \pi /11+cos7 \pi /11+cos9 \pi /11=?

1
by stark1

• Brainly User
2014-10-04T11:56:04+05:30
Multiplying the given expression by Sin(5π/11),

Sin(5π/11)Cos(π/11) + Sin(5π/11)Cos(3π/11) + Sin(5π/11)Cos(5π/11) +
Sin(5π/11)Cos(7π/11) + Sin(5π/11)Cos(9π/11)

Use the formula SinC SinD = (1/2)[Sin(C + D) + Sin(C - D)

= (1/2) [ {Sin(6π/11) + Sin(4π/11)} + {Sin(8π/11) + Sin(2π/11)}
+ {Sin(10π/11) + Sin(0)} + {Sin(12π/11) + Sin(-2π/11)}
+ {Sin(14π/11) + Sin(-4π/11)]
=  (1/2) [ {Sin(6π/11) + Sin(4π/11)} + {Sin(8π/11) + Sin(2π/11)}
+ {Sin(10π/11)} + {Sin(-10π/11) + Sin(-2π/11)}
+ {Sin(-8π/11) + Sin(-4π/11)]

=   (1/2) [ Sin(6π/11) + {Sin(4π/11) + Sin(-4π/11)}+ {Sin(8π/11) + Sin(-8π/11)}
+ {Sin(2π/11) + Sin(-2π/11)} + {Sin(10π/11) + Sin(-10π/11)}]

= (1/2) x [Sin(6π/11)]
AS Sin A = Sin(π - A),
= (1/2) x {Sin(5π/11)} -------------------------(1)

Since we had multiplied the given expression by Sin(5π/11), we divide by the same now,

The original given expression = [(1/2) x {Sin(5π/1)}] / {Sin(5π/11)}
= 1/2