Prove that `sin alpha+sin(alpha+2pi/3)+sin(alpha+4pi/3)=0`

Solve sin 3alpha=4 sin alpha sin (x+alpha)sin(x-alpha) where alpha=n pi,n in Z .

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

if tanalpha , tanbeta are the roots of x^2+px+q=0 ,show that sin^2(alpha+beta)+p sin(alpha+beta)cos(alpha+beta)+q cos^2(alpha+beta)=q

Prove that. cos(pi/3)cos(pi/6)+sin(pi/3)sin(pi/6)=cos(pi/6)

If A =[(cos alpha, sin alpha),(-sin alpha, cos alpha)] , then find A^(T) A.

If A=[(0, -tan""(alpha)/(2)),(tan""(alpha)/(2),0)] , then show that (I+A) =(I-A) [(cos alpha, - sin alpha),(sin alpha, cos alpha)] , where I is the identity matrix of order 2.

Prove that sin^2(pi/18)+sin^2(pi/9)+sin^2(7pi/18)+sin^2(4pi/9)=2

If A=[(cos alpha, -sin alpha),(sin alpha, cos alpha)] and A+A^(T) =l , write down the general values of alpha .

If xsintheta=ysin(theta+(2pi)/3)=z sin (theta+(4pi)/3), ,prove that xy + yz + zx = 0.

Prove that the locus of the points of intersection of the lines x cos alpha + y sin alpha = a and x sin alpha - y cos albha = b is a cicle, whatever may be alpha