Prove that
`|2 alpha+beta+gamma+delta alphabeta+gammadelta`
`alpha+beta+gamma+delta 2(alpha+beta)(gamma+delta) alphabeta(gamma+delta)+gammadelta(alpha+beta)`
`alphabeta+gammadeltaalphabeta(gamma+delta)+gammadelta(alpha+beta)2alphabetagammadelta|=0`

If cot alpha=tan (beta+delta) then alpha+beta+delta =

Prove that |((beta+gamma-alpha-delta)^4,(beta+gamma-alpha-delta)^2,1),((gamma+alpha-beta-delta)^4,(gamma+alpha-beta-delta)^2,1),((alpha+beta-gamma-delta)^4,(alpha+beta-gamma-delta)^2,1)|= -64(alpha-beta)(alpha-gamma)(alpha-delta)(beta-delta)(gamma-delta)(gamma-beta)

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)

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

Evaluate: |[alpha, beta,gamma],[alpha^2,beta^2,gamma^2],[beta+gamma,gamma+alpha,alpha+beta]|

Prove that |{:(betagamma,betagamma'+beta'gamma,beta'gamma'),(gammaalpha,gammaalpha'+gamma'alpha,gamma'alpha'),(alphabeta,alphabeta'+alpha'beta,alpha'beta'):}| =(alphabeta'-alpha'beta)(betagamma'-beta'gamma)(gammaalpha'-gamma'alpha) .

If sec alpha = sec beta sec gamma+ tan beta tan gamma , then show that sec beta= sec gamma sec alpha+- tan gamma tan alpha .

If alpha+ beta=(pi)/(2) and beta+ gamma= alpha then prove that tan alpha=tan beta+2 tan gamma .

Simplify: cot (beta - gamma) cot (gamma - alpha) + cot (gamma - alpha) cot (alpha - beta) + cot (alpha - beta) cot (beta - gamma)

If alpha, beta, gamma ,are positive acute angles and sin (alpha+beta-gamma)=cos(beta+gamma-alpha)=tan (gamma+alpha-beta)=1 find alpha,beta and gamma .