If ` sin alpha =(1)/(2) ,` prove that `(3 cos alpha - 4 cos ^(3) alpha )=0.`

If alpha = (pi)/(13), Prove that cos alpha cos 2alpha cos 3alpha cos 4alpha cos 5alpha cos 6alpha = (1)/(64) .

If alpha= (2pi)/(7) , then prove that cos alpha+ cos 2 alpha+ cos 4alpha=-(1)/(2) . Hence, deduce that sin alpha+ sin 2alpha+ sin 4alpha=(sqrt(7))/(2)

If y = (2 sin alpha )/(1 + cos alpha + sin alpha) then, prove that (1 - cos alpha + sin alpha)/(1 + sin alpha) = y .

If cos (alpha - beta) + cos (beta - gamma) + cos (gamma - alpha) = (-3)/(2) then prove that cos alpha + cos beta+ cos gamma = sin alpha + sin beta + sin gamma = 0 .

For alpha , beta in R , prove that (cos alpha + cos beta )^(2)+ (sin alpha + sin beta )^(2)=4cos^(2). ((alpha-beta))/(2)

(( cos alpha - cos 3 alpha ) ( sin 8 alpha + sin 2 alpha ))/( ( sin 5 alpha - sin alpha ) ( cos 4 alpha - cos 6 alpha ))=

If (cos^(4)alpha)/(cos^(2) beta) + (sin^(4)alpha)/(sin^(2)beta) = 1, prove that (cos^(4)beta)/(cos^(2) alpha) + (sin^(4)beta)/(sin^(2)alpha)= 1

If (cos^4 alpha)/(cos^2 beta)+ (sin^4 alpha)/(sin^2 beta) =1 , prove that : (cos^4 beta)/(cos^2 alpha)+ (sin^4 beta)/ (sin^2 alpha) =1 .

If alpha = pi/3 , prove that cos alpha*cos 2alpha*cos 3alpha*cos 4alpha*cos 5alpha*cos 6alpha = -1/16 .

If alpha = pi/3 , prove that cos alpha*cos 2alpha*cos 3alpha*cos 4alpha*cos 5alpha*cos 6alpha = -1/16 .