Question

Show that sin 10 + sin 20 + sin 40 + sin 50 = sin 70 + sin 80

Answer 1

Look here!!!

sin 10° + sin20° + sin40° + sin50° = sin70° + sin80°
we will first solve

L.H.S. = 2sin15cos5+2sin45cos5 [ using sin C+sin D= 2sin C+D/2 cos C-D/2 for sin10+sin20 & sin40+sin50]

           = 2cos5 (sin15+sin45)

           = 2cos5 (2sin30cos15) [ using sin C+sin D= 2sin C+D/2 cos C-D/2 ]

           = 2cos5 (2 x 1/2 x cos15)

           = 2cos5 cos15

 
Now its the turn for RHS

R.H.S. = sin70+sin80

           = 2sin75cos5 [ using sin C+sin D= 2sin C+D/2 cos C-D/2 ]

           

sin75 = sin(90-15) = cos 15

 

L.H.S = 2cos5 cos15

R.H.S. = 2cos15 cos5 [ since, sin75 = cos15 ]

Got it

Answer 2

Answer:

refer to the attachment for the proof.

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