Question

prove that sinx+ sin3x+sin5x+sin7x=4sin4xcos2xcosx​

Answer 1

Step-by-step explanation:

sinx+sin3x+sin5x+sin7x

sinx+7x+sin3x+sin5x

[sin(A+B)]sinC+sinD=2sin

2

C+D

cos

2

C−D

Now,

2sin(

2

x+7x

)cos(

2

x−7x

)+2sin(

2

3x+5x

)cos(

2

3x−5x

)

2sin4xcos3x+2sin4xcosx

2sin4x(cos3x+cosx)

2sin4x[2cos

2

C+D

cos

2

C−D

]

2sin4x[2cos

2

3x+x

cos

2

3x−x

]

2sin4x×2cos2xcosx

4sin4xcos2xcosx.