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Question

If x+y+z= π2, then show that, sin2x+sin2y+sin2z=4cosxcosycosz.

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Solution

We have x+y+z=π2, then to prove

sin2x+sin2y+sin2z=4cosxcosycosz

L.H.S

=sin2x+sin2y+sin2z

=2sin(x+y)cos(xy)+2sinzcosz

=2sin(π2z)cos(xy)+2sinzcosz

=2cosz[cos(xy)+sinz]


=2cosz(cos(xy)+sin(π2(x+y)))

=2cosz(cos(xy)+cos(x+y))

=2cosz2cosxcosy

=4cosxcosycosz

=RHS
Hence proved

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