prove that `cosalpha\ cos2alpha\ cos4alpha......cos(2^(n-1)alpha)=(sin2^nalpha)/(2^n sinalpha)\ for\ a l l\ n in N`

If n-1,2,3,dot,t h e ncosalpha"cos"2alpha"cos"4alphacos2^(n-1)alphai se q u a lto (sin2nalpha)/(2nsinalpha) (b) (sin2^nalpha)/(2^n"sin"2^("n"-1)alpha) (c) (sin4^(n-1)alpha)/(4^(n-1)sinalpha) (d) (sin2^nalpha)/(2^n"sin"alpha)

1/(cosalpha+cos3alpha)+1/(cosalpha+cos5alpha)....1/(cosalpha+cos(2n+1)alpha

Prove that : cos^2alpha+cos^2(alpha+beta)-2cosalphacosbetacos(alpha+beta)=sin^2beta

If A=[cosalpha+sinalpha,sqrt(2)sinalpha; -sqrt(2)sinalpha,cosalpha-sinalpha] , prove that A^n=[cosn\ alpha+sinn\ alpha, sqrt(2)sinn\ alpha; -sqrt(2)sinn\ alpha, cosn\ alpha-sinn\ alpha] for all n in Ndot

Prove that: cos2alpha\ cos2beta+sin^2(alpha-beta)-sin^2(alpha+beta)=cos2(alpha+beta) .

The value of tan alpha +2tan(2alpha)+4tan(4alpha)+...+2^(n-1)tan(2^(n-1)alpha)+2^ncot(2^nalpha) is

Prove that (sinalpha)/(1+cosalpha)+(1+cosalpha)/(sin alpha)=2"cosec "alpha .

If 4nalpha =pi then cot alpha cot 2 alpha cot 3alpha ...cot (2n-1)alpha n in Z is equal to

One side of a square makes an angle alpha with x axis and one vertex of the square is at origin. Prove that the equations of its diagonals are x(sin alpha+ cos alpha) =y (cosalpha-sinalpha) or x(cos alpha-sin alpha) + y (sin alpha + cos alpha) = a , where a is the length of the side of the square.

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