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prove that : cos 4 180/16 +cos 4 540/16 + cos 4 900/16 + cos 4 1260/16 = 3/2

prove that : cos4180/16 +cos4540/16 + cos4900/16 + cos41260/16 = 3/2
 

Grade:11

1 Answers

Akshay
185 Points
8 years ago
write cos(1260/16) = cos(90-180/16) = sin(180/16), 
and cos(900/16) = cos(90-540/16) = sin(540/16),
you will get : cos4(180/16) + sin4(180/16) + cos4(540/16) + sin4(540/16)
By applying a2+b2 = (a+b)2 – 2ab, you will get
= (cos2(180/16) + sin2(180/16))2 – 2*cos2(180/16)*sin2(180/16) + (cos2(540/16) + sin2(540/16))2 – 2*cos2(540/16)*sin2(540/16),
= 2 – 2*( cos2(540/16)*sin2(540/16) + cos2(180/16)*sin2(180/16) ),
use sinx*cosx= ½ * sin2x,
=2 – ½ * ( sin2(1080/16) + sin2(360/16) ),
write sin(1080/16) = sin(90 – 360/16) = cos(360/16),
= 2 – ½ * ( cos2(360/16) + sin2(360/16) ),
= 2- ½ 
= 3/2 

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