Question

sin10+sin50+sin20+sin40

Answer 1

sin50+sin10+sin40+sin20
sinA+sinB=2sin(a+b)/2.cos(a-b)/2
2sin(50+10)/2.cos(50-10)/2+2sin(40+20)/2.cos(40-20)/2
2sin60/2 cos40/2+2sin60/2.cos20/2
2sin30cos20+2sin30cos10
2*1/2 cos20 + 2*1/2 cos10
cos20+cos10
cosA+cosB=2cos(a+b)/2.cos(a-b)/2
2cos(20+10)/2.cos(20-10)/2]
2cos30/2.cos10/2
2cos15.cos5
2sin(90-15)=2sin75.cos5
2sin(45+30).cos5
2(sin45cos30+cos45.sin30)cos5
2(1/√2*√3/2+1/√2.1/2)cos5
2/√2(√3/2+1/2)cos5
√2(√3+1)/2 cos5
(√3+1)/√2 cos 5
cos 5 is tends to 1 so take cos 5 = 1 
because cos is a decrasing function so value of cos 5 is 0.99872something ≈1
(√3+1)/√2 *1
ans will be (√3+1)/√2