Show that the lines `(x-a+d)/(alpha-delta) = (y-a)/alpha = (z-a-d)/(alpha+delta)` and `(x-b+c)/(beta-gamma) = (y-b)/beta = (z-b-c)/(beta+gamma)` are coplanar.

If cos (alpha+beta) sin (gamma+delta)=cos (alpha-beta) sin (gamma-delta) , prove that : cot alpha cot beta cot gamma= cot delta .

If cosalpha+cosbeta+cosgamma=0=sinalpha+sinbeta+singamma , then which of the following is/are true:- (a) cos(alpha-beta)+cos(beta-gamma)+cos(gamma-delta)=-3/2 (b) cos(alpha-beta)+cos(beta-gamma)+cos(gamma-delta)=-1/2 (c) sumcos2alpha+2cos(alpha+beta)+2cos(beta+gamma)+2cos(gamma+alpha)=0 (d) sumsin2alpha+2sin(alpha+beta)+2sin(beta+gamma)+2sin(gamma+alpha)=0

If alpha, beta, gamma are the roots of the cubic x^(3)-px^(2)+qx-r=0 Find the equations whose roots are (i) beta gamma +1/(alpha), gamma alpha+1/(beta), alpha beta+1/(gamma) (ii) (beta+gamma-alpha),(gamma+alpha-beta),(alpha+beta-gamma) Also find the valueof (beta+gamma-alpha)(gamma+alpha-beta)(alpha+beta-gamma)

If (tan (theta+alpha))/a= (tan (theta+beta))/b= (tan (theta+ gamma))/c , prove that : (a+b)/(a-b) sin^2 (alpha- beta) +(b+c)/(b-c) sin^2 (beta-gamma)+ (c+a)/(c-a) sin^2 (gamma-alpha)=0 .

If alpha, beta are the roots of the quadratic equation x^2 + bx - c = 0 , the equation whose roots are b and c , is a. x^(2)+alpha x- beta=0 b. x^(2)-[(alpha +beta)+alpha beta]x-alpha beta( alpha+beta)=0 c. x^(2)+[(alpha + beta)+alpha beta]x+alpha beta(alpha + beta)=0 d. x^(2)+[(alpha +beta)+alpha beta)]x -alpha beta(alpha +beta)=0

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|

If alpha,beta are the roots of x^2+p x+q=0a n dgamma,delta are the roots of x^2+r x+s=0, evaluate (alpha-gamma)(alpha-delta)(beta-gamma)(beta-delta) in lterms of p ,q ,r ,a n dsdot Deduce the condition that the equation has a common root.

If 2 sin alphacos beta sin gamma=sinbeta sin(alpha+gamma),then tan alpha,tan beta and tan gamma are in