Without expanding show that `|(0,sinalpha, -cosalpha),(-sinalpha,0,sinbeta),(cosalpha, -sinbeta,0)| = 0`

Evaluate triangle = |[0,sinalpha,-cosalpha],[-sinalpha,0,sinbeta],[cosalpha,-sinbeta,0]|

Prove that: (cosalpha+cosbeta)^2+(sinalpha+sinbeta)^2=4cos^2((alpha-beta)/2)

If A = {:[(cosalpha, sinalpha),(-sinalpha, cosalpha)] , then find A^2

Show that the matrix: A = {:[(cosalpha, -sinalpha),(sinalpha,cosalpha)] is orthogonal.

Using properties of determinants, prove that: |[sinalpha,cosalpha,cos(alpha+delta)],[sinbeta,cosbeta,cos(beta+delta)],[singamma,cosgamma,cos(gamma+delta)]| = 0

Without expanding , show that |(a,b,c),(a+2x,b+2y,c+2z),(x,y,z)|= 0 .

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.

Show taht 2tan^-1(tan(alpha/2)tan(pi/4-beta/2))=tan^-1((sinalphacosbeta)/(cosalpha+sinbeta))