Answer:

Answer:

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\begin{gathered}\begin{gathered}1. \: \: \sin(2 \alpha ) = 2 \sin( \alpha ) \COS( \alpha ) \\\end{gathered} 1.sin(2α)=2sin(α)cos(α)\end{gathered}1.sin(2α)=2sin(α)cos(α)1.sin(2α)=2sin(α)cos(α)

2. \: \: \sin(180 - \alpha ) = \sin( \alpha )2.sin(180−α)=sin(α)2.sin(180−α)=sin(α)2.sin(180−α)=sin(α)

3. \: \: \sin(90 - \alpha ) = \cos( \alpha )3.sin(90−α)=cos(α)3.sin(90−α)=cos(α)3.sin(90−α)=cos(α)

L.H.S :- Sin10. SIN30. Sin50. Sin70L.H.S:−Sin10.Sin30.Sin50.Sin70

\begin{gathered}\begin{gathered}= \cos(90 - 10) \sin(30) \cos(90 - 40) \cos(90 - 70) \\ = \cos(80) \: \frac{1}{2} \: \cos(40) \: \cos(20) \\ = \frac{1}{4 \sin(20) } \cos(8 0 ) \cos(40) . \: 2 \sin(20) \cos(20) \\ = \frac{1}{8 \sin(20) } \cos(80) 2. \cos(40) \sin(40) \\ = \frac{1}{16 \sin(20) } 2 \cos(80) \sin(80 ) \\ = \frac{1}{16 \sin(20) } \sin(160) \\ = \frac{1}{16 \sin(20) } \sin(180 - 20) \\ = \frac{1}{16 \sin(20) } \sin(20) \\ = \frac{1}{16} \: \: \: \: \: \: \: \: .........R.H.S\end{gathered}\end{gathered}=cos(90−10)sin(30)cos(90−40)cos(90−70)=cos(80)21cos(40)cos(20)=4sin(20)1cos(80)cos(40).2sin(20)cos(20)=8sin(20)1cos(80)2.cos(40)sin(40)=16sin(20)12cos(80)sin(80)=16sin(20)1sin(160)=16sin(20)1sin(180−20)=16sin(20)1sin(20)=161.........R.H.S

=cos(90−10)sin(30)cos(90−40)cos(90−70)=cos(90−10)sin(30)cos(90−40)cos(90−70)

=cos(80) 2/1=cos(80)2/1

cos(40)cos(20)= 4sin(20)1cos(40)cos(20)=4sin(20)1

cos(80)cos(40).2sin(20)cos(20)= 8sin(20)1cos(80)cos(40).2sin(20)cos(20)=8sin(20)1

cos(80)2.cos(40)sin(40)= 16sin(20)1cos(80)2.cos(40)sin(40)=16sin(20)1

2cos(80)sin(80)= 16sin(20)12cos(80)sin(80)=16sin(20)1

sin(160)= 16sin(20)1sin(160)=16sin(20)1

sin(180−20)= 16sin(20)1sin(180−20)=16sin(20)1

sin(20)= 16/1sin(20)=16/1

.........R.H.S