Show that `4 sin alpha.sin (alpha + pi/3) sin (alpha + 2pi/3) = sin 3alpha`

show that : sin^2 alpha + sin^2(alpha +( 2pi)/n) +sin^2(alpha +( 4pi)/n) + ..... n terms = n/2

Prove that sinalpha*sin(60-alpha)sin(60+alpha) = 1/4*sin3alpha

If sin 3 alpha =4 sin alpha sin (x+alpha ) sin(x-alpha ) , then

(cos ^(3) alpha - cos 3 alpha )/( cos alpha ) + (sin ^(3)alpha + sin 3 alpha )/( sin alpha ) = 3.

Evaluate : |{:( 0, sin alpha , - cos alpha ), ( - sin alpha , 0 , sin beta) , (cos alpha , -sin beta , 0):}|

Show that (sin (alpha + beta))/( sin (alpha + beta)) = 2, given that tan alpha = 2 tan beta.

Sum the infinite series : sinalpha+x sin (alpha+beta)+ x^2/(2!)sin(alpha+2beta)+x^3/(3!) sin (alpha+3beta)+..

Show that: sin^2 alpha + sin^2 beta + 2sinalpha sinbeta cos(alpha+beta)=sin^2 (alpha+beta)

Prove that 2 sin^2 beta + 4 cos(alpha + beta) sin alpha sin beta + cos 2(alpha + beta) = cos 2alpha

If sin(x +3alpha)=3 sin (alpha-x) , then