यदि `A + B + C = 180^(@)` है , तो `sin 2A - sin 2B - sin 2C` किसके बराबर है ?

Options

A. `- 4 sin A^(~) sin B sin C`
B. `- 4 cos sin A sin B cos C`
C. `-4 cos A cos B sin C`
D. `-4 sinA cos B cos C`

Explanation

दिया है ,
`sin 2 A - sin2B - sin 2C`
`= sin 2A - [ sin 2B + sin 2C]`
`= sin 2A - [2 sin + sin 2C]`
`= sin 2A - [2 sin ((2B + 2C)/(2)) cos ((2B - 2C)/(2))]`
`[ :' Sin C + Sin D = 2 Sin' (C+D)/(2) Cos'(C-D)/(2)]`
`2 Sin A CosA - [2Sin (B+C) Cos (B-C)]`
`[ :' sin 2A = 2 sin A cos A]`
`= 2 sin A cos A - [2sin (180-A)cos (B-C)]`
`= 2 sin A cos A - 2 sin A cos (B-C)`
`[ :' sin (180-theta) = sin theta]`
`= 2 sin A [cos A - cos (B-C)]`
`= 2 sin A [cos (180-(B +C) - cos (B-C)]`
`= 2 sin A [-cos (B +C) - cos (B-C)]`
`= - 2 sin A [ cos (B + C) - cos (B-C)]`
`= - 2 sin A [2 cos B cos C]`
`:' cos C + cos D`
`= 2 cos ((C + D)/(2)) cos ((C-D)/(2))`
`= - 4 sin A cos B cos C`