Prove that, sin10°+sin20°+sin40°+sin50°= sin70°+sin80°
Solution:
L.H.S = sin10°+sin20°+sin40°+sin50°
= sin10°+sin50° +sin20°+sin40°
= 2 sin$\frac{10° + 50°}{2}$ cos \frac{10° – 50°}{2} + 2 sin $\frac{20° + 40°}{2}$ cos $\frac{20° -40°}{2}$
= 2 sin30° cos(-20°) + 2 sin30° cos(-10°)
= 2× $\frac{1}{2}$ cos20° +2×$\frac{1}{2}$ cos10°
= cos20° + cos10°
= cos(90°-70°)+ cos(90°-80°)
= sin70°+ sin80° = R.H.S [Proved]