If x+y+z=π/2, then prove that: sin 2x + sin 2y + sin 2z= 4 cos x cos y cos z?

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Answer 1 · 2018-06-14T06:51:01

Kindly refer to the Explanation.

Given that, #x+y+z=pi/2#, we have,

#x+y=pi/2-z.....(ast_1), &, z=pi/2-(x+y)......(ast_2)#.

Now, #ul(sin2x+sin2y)+sin2z#,

#=2sin((2x+2y)/2)cos((2x-2y)/2)+sin2z#,

#=2sin(x+y)cos(x-y)+sin2z#,

#=2sin(pi/2-z)cos(x-y)+ul(sin2z).......................[because, (ast_1)]#,

#=2coszcos(x-y)+ul(2sinzcosz)#,

#=2cosz{cos(x-y)+sinz}#,

#=2cosz{cos(x-y)+sin(pi/2-(x+y))}......[because, (ast_2)]#,

#=2cosz{cos(x-y)+cos(x+y)}#,

#=2cosz*2cos(((x+y)+(x-y))/2)cos(((x+y)-(x-y))/2),#

#=4coszcosxcosy#, as desired!

Enjoy Maths.!