2009/031) simplify sin10*sin50*sin60*sin70*sin90

we know sin 90 = 1 and sin 60 = sqrt(3)/2
hence
sin10*sin50*sin60*sin70*sin90 = sqrt(3/2) sin 10 sin 50 sin 70
= sqrt(3)/2 sin 10 sin (90-40) sin (90-20)
= sqrt(3)/2 sin 10 cos 40 cos 20 ( as sin (90-x) = cos x)
= sqrt(3)/2 sin 10 cos 20 cos 40 ..1

Knowing sin 2a =2 sin a cos a and 20 is double of 10 and 40 is double od 20 we proceed

now sin10 cos 20 cos 40 = (cos 10 sin 10 cos 20 cos 40)/cos 10
= (2 cos 10 sin 10 cos 20 cos 40)/(2 cos 10)
= ( sin 20 cos 20 cos 40)/( 2 cos 10
= (2 sin 20 cos 20 cos 40)/( 4 cos 10)
= ( sin 40 cos 40)/( 4 cos 10)
= ( 2 sin 40 cos 40)/( 8 cos 10)
= sin 80 /( 8 cos 10)
= cos 10/(8 cos 10) = 1/8 ...2

from 1 and 2 we get
sin10*sin50*sin60*sin70*sin90 = sqrt(3)/16