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Question

Show that sin10+sin20+sin40+sin50=sin70+sin80.

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Solution

Let x=sin100+sin200+sin400+sin500=sin100+sin500+sin200+sin400

Applying trigonometric identity

sina0+sinb0 =2sin(a0+b02)cos(a0b02)


x==2sin(100+5002)cos(2005002)+2sin(200+4002)cos(2004002)

=2sin300cos(200)+2sin300cos(100)

=2×12cos200+2×12cos100=cos200+cos100

=cos(900200)+cos(900100)=sin700+sin800
sin100+sin200+sin400+sin500=sin700+sin800

Hence Proved.

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