Question

prove that Sin10°+Sin20°+Sin40°+Sin50°=Sin70°+Sin80°

Answer 1

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

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 ]

Answer 2

sin10+sin40+sin50+sin20=sin70+ sin80
sin A + sin B = 2 sin ½ (A + B) cos ½ (A − B)

so sin 50 + sin 10 = 2 sin 30 cos 20 = 2 *1/2 sin (90-20) = sin 70

and sin 40 + sin 20 = 2 sin 30 cos 10 = 2 *1/2 sin (90-80) = sin 80

Hence proved