Proving \frac{2 \sin x+\sin 2x}{2 \sin x-\sin 2x}=\csc^2

Arthur Pratt

Arthur Pratt

Answered question

2021-12-30

Proving 2sinx+sin2x2sinxsin2x=csc2x+2cscxcotx+cot2x
Proving right hand side to left hand side:
csc2x+2cscxcotx+cot2x=1sin2x+2cosxsin2x+cos2xsin2x (1)
=cos2x+2cosx+1sin2x (2)
=1sin2x+1+sin2xsinxsin2x (3)
=2sinxsin3x+sin2xsinxsin2x (4)
=2sinxsin3x+sin2xsin3x (5)
I could not prove further to the left hand side from here.

Answer & Explanation

hysgubwyri3

hysgubwyri3

Beginner2021-12-31Added 43 answers

We have that, using sin2x=2sinxcosx and cancelling out 2sinx0
2sinx+sin2x2sinxsin2x=2sinx+2sinxcosx2sinx2sinxcosx=1+cosx1cosx
then, assuming cosx1 ( which holds since we need sinx0 for the original expression), multiply by 1+cosx1+cosx and simplify to obtain the result using that 1cos2x=sin2x
Cheryl King

Cheryl King

Beginner2022-01-01Added 36 answers

The following relations would be used in the answer:
1cos2x=sin2x
sin2x=2sinxcosx
Taking LHS,
2sinx+sin2x2sinxsin2x=2sinx(1+cosx)2sinx(1cosx)
Cancelling sin2x
=1+cosx1cosx
Multiplying 1+cosx to numerator and denominator
=(1+cosx)21cos2x
which is equivalent to
=(1+cos2x+2cosx)sin2x
Divinding each term by denominator, we get LHS as
=csc2x+2cscxcotx+cot2x
which is the RHS.
Hence, Proved.
nick1337

nick1337

Expert2022-01-08Added 777 answers

2sinx+sin2x2sinxsin2x=2sinx+2sinxcosx2sinx2sinxcosx=1+cosx1cosx=(1+cosx)21cos2x=1+2cosx+cos2xsin2x=csc2x+2cscxcotx+cot2x

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