Prove that `sum_(alpha+beta+gamma = 10) (10 !)/(alpha!beta!gamma!)=3^(10)dot`

"Prove that : " sin alpha + sin beta + sin gamma - sin ( alpha + beta + gamma) =4 "sin" (alpha+beta)/2 "sin"(beta + gamma)/2 "sin"(gamma+alpha)/2

If alpha, beta, gamma are positive acute angles, prove that sin alpha+sin beta+ sin gamma gt sin (alpha+ beta+ gamma)

For all values of alpha,beta,gamma prove that: cos alpha + cos beta+ cos gamma+ cos(alpha+beta+gamma) ="4cos"(alpha+beta)/2cdot"cos" (beta+gamma)/2cdot"cos"(gamma+alpha)/2

If alpha,beta,gamma, in (0,pi/2) , then prove that (sin(alpha+beta+gamma))/(sinalpha+sinbeta+singamma)<1

Let x-ysin alpha-zsin beta=0 , xsin alpha+zsin gamma-y=0 and xsin beta+ysin gamma-z=0 be the equations of the planes such that alpha+beta+gamma=pi//2 (where alpha,beta and gamma!=0)dot Then show that there is a common line of intersection of the three given planes.

Prove that, |{:(alpha,beta,gamma),(alpha^2,beta^2,gamma^2),(beta+gamma,gamma+alpha,alpha+beta):}|=(alpha-beta)(beta-gamma)(gamma-alpha)(alpha+beta+gamma)

Prove that cosalpha+cosbeta+cosgamma+cos(alpha+beta+gamma) = 4cos((alpha+beta)/2)cos((beta+gamma)/2)cos((gamma+beta)/2)

if 0ltalpha,beta,gamma lt(pi/2) prove that sin alpha+sin beta+sin gamma gtsin(alpha+beta+gamma)

If (tan (alpha- beta+ gamma))/(tan (alpha+ beta- gamma))=(tan beta)/( tan gamma) , show that either sin (beta- gamma)=0 or sin 2 alpha+ sin 2 beta+ sin 2 gamma=0

If alpha,beta,gamma are different from 1 and are the roots of a x^3+b x^2+c x+d=0a n d(beta-gamma)(gamma-alpha)(alpha-beta)=(25)/2 , then prove that |alpha/(1-alpha)beta/(1-beta)gamma/(1-gamma)alphabetagammaalpha^2beta^2gamma^2|=(25 d)/(2(a+b+c+d))