Show that the determinant `Delta(x)` given by `Delta(x)` =
`|{:(sin(x+alpha),cos(x+alpha),a+xsinalpha),(sin(x+beta),cos(x+beta),b+xsinbeta),(sin(x+gamma),cos(x+gamma),c+xsingamma):}|` is independent of `x`.

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

If A = [(cos alpha, sin alpha),(sin alpha, cos alpha)] and B = [(cos beta, sin beta),(sin beta, cos beta)] show that AB = BA

Prove by vectors that : cos (alpha + beta) = cos alpha cos beta- sin alpha sin beta .

Prove by vectors that : cos (alpha - beta) = cos alpha cos beta+ sin alpha sin beta .

Without expanding, show that the following determinants vanish: {:|(1,cos(beta-alpha),cos(gamma-alpha)),(cos(alpha-beta),1,cos(gamma-beta)),(cos(alpha-gamma)),cos(beta-gamma),1|

The value of the determinant |{:(1,sin(alpha-beta)theta,cos (alpha-beta)theta),(a, sinalphatheta,cos alphatheta),(a^(2),sin(alpha-beta)theta,cos(alpha-beta)theta):}| is independent of (a) alpha (B) beta (C) theta (D)a

Find the value of f(alpha, beta) =|(cos (alpha + beta), -sin (alpha + beta), cos 2beta),(sin alpha, cos alpha, sin beta),(-cos alpha, sin alpha, cos beta)|