Question

prove that sin 10 + sin 20 + sin 40 + sin 50=sin 70 + sin 80​

Answer 1

Answer:

sin10+sin40+sin50+sin20=sin70+ sin80

Here we will use the formula

sinA+sinB=2sin(A+B/2)cos(A-B/2)

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+20/2)cos(40-20/2)

=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

Hence proved