Show that `|{:(-1+xsinalpha,cos(x+alpha),sin(x+alpha)),(13+x sin beta,cos(x+beta),sin(x+beta)),(-12+x sin gamma,cos (x+gamma),sin(x+gamma)):}|` is independent of x.

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.

Prove that det [[sin alpha, cos alpha, sin (alpha + delta) sin beta, cos beta, sin (beta + delta) sin gamma, cos gamma, sin (gamma + delta)]] = 0

sin alpha, cos alpha, cos (alpha + delta) sin beta, cos beta, cos (beta + delta) sin gamma, cos gamma, cos (gamma + delta)] | = 0

If Delta=|(sin alpha, cos alpha, sin alpha+cos beta),(sin beta, cos alpha, sin beta+cos beta),(sin gamma, cos alpha, sin gamma+cos beta)| then Delta equals

Suppose alpha ,beta , gamma in R are such that sin alpha, sin beta , sin gamma ne 0 and Delta = |{:(sin^(2) alpha , sin alpha cos alpha , cos^(2) alpha),(sin^(2) beta , sin beta cos beta , cos^(2) beta),(sin^(2) gamma , sin gamma cos gamma , cos^(2) gamma):}| then Delta cannot exceed

Without expanding evaluate the determinant det[[sin alpha,cos alpha sin(alpha+delta)sin beta,cos beta,sin(beta+delta)sin gamma,cos gamma,sin(gamma+delta)]]

If Delta = det [[sin alpha, cos alpha, sin alpha + cos alphasin beta, cos alpha, sin beta + cos betasin gamma, cos alpha, sin gamma + cos beta]] then Delta

Without expanding evaluate the determinant |sin alpha cos alpha sin(alpha+delta)sin beta cos beta sin(beta+delta)sin gamma cos gamma sin(gamma+delta)|

The orthocentre of triangle formed by (a cos alpha,a sin alpha),(a cos beta,a sin beta),(a cos gamma,a sin gamma) is

Delta ABC has vertices A(a cos alpha,a sin alpha),B(a cos beta,a sin beta) and C(a cos gamma,a sin gamma) then its orthocentre is