We are going to prove it like this:
Lets use the formula sinA+sinB=2sin(A+B/2)cos(A-B/2)
Now we are going to take the left side of equation
sin10+sin40+sin50+sin20
Arranging
=(sin50+sin10)+(sin40+sin20)
Applying the above formula.
=2sin(50+10/2)cos(50-10/2)+2sin(40+20/…
=2sin(30)cos(20)+2sin(30)cos(10)
=2sin30{cos20+cos10}
Again using the formula
cosA+cosB= 2cos(A+B/2)cos(A-B/2)
=2sin30{2cos(20+10/2).cos(20-10/2)}
=2sin30{2cos(15).cos(5)}
=2(1/2){2cos15.cos5} as sin30=1/2
=2cos15.cos5
Taking right side of equation
sin70+ sin80
Using the formula
sinA+sinB
= 2sin(A+B/2)cos(A-B/2)
=2sin(70+80/2)cos(70-80/2)
=2sin75cos5
=2sin(90-15)cos5
=2cos15.cos5
Hope this helps
