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

1 Answer
Jun 14, 2018

Kindly refer to the Explanation.

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.!