Show that the determinant `Delta` (x) is 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.

If sin 3 alpha =4 sin alpha sin (x+alpha ) sin(x-alpha ) , then

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

If f(x) = |{:(cos (x+alpha),cos(x+beta),cos(x+gamma)),(sin (x+alpha),sin(x+beta),sin(x+gamma)),(sin(beta+gamma),sin(gamma+alpha),sin(alpha+beta)):}| then f(theta)-2f(phi)+f(psi) is equal to

y = e^(x)(a cos x + b sin x)

The value of the determinant |(1,(alpha^(2x)-alpha^(-2x))^2,(alpha^(2x)+alpha^(-2x))^2),(1,(beta^(2x)-beta^(-2x))^2,(beta^(2x)+beta^(-2x))^2),(1,(gamma^(2x)-gamma^(-2x))^2,(gamma^(2x)+gamma^(-2x))^2)| is (a) 0 (b) (alphabeta gamma)^(2x) (c) (alpha beta gamma)^(-2x) (d) None of these

The angle between the lines (x^(2)+y^(2))sin^(2)alpha=(x cos beta-y sin beta)^(2) is

Prove that (cos(9x)-cos(5x))/(sin(17x)-sin(3x))=-(sin(2x))/(cos(10x))

cos(2pi -x) cos (-x) - sin(2pi +x) sin (-x) = 0