Question

Prove that: `sin(pi/10) sin((13pi)/10)=-1/4`

Answer 1

Here, we will use,
`sin 18^@ = (sqrt5-1)/4 and cos 36^@ = (sqrt5+1)/4`
Now, `L.H.S. = sin(pi/10)sin((13pi)/10)`
`=sin18^@sin(pi+(3pi)/10)`
`=sin18^@(-sin((3pi)/10))`
`=-sin18^@sin54^@`
`=-sin18^@sin(90^@-36^@)`
`=-sin18^@cos36^@`
`=-((sqrt5-1)/4)( (sqrt5+1)/4) `
`=-((5-1)/16)`
`=-4/16`
`=-1/4 = R.H.S.`