If `cot alpha cot beta=2`, show that `(cos (alpha+beta))/(cos (alpha-beta))=(1)/(3)`

If cos alpha+cos beta=1/3 and sin alpha+sin beta=1/4 ,show that tan((alpha+beta)/2)=3/4

The distance between the points (a cos alpha, a sin alpha) and (a cos beta, a sin beta) is a) 2|sin((alpha-beta)/(2))| b) 2|a sin((alpha-beta)/(2))| c) 2|a cos((alpha-beta)/(2))| d) |a cos((alpha-beta)/(2))|

If cosalpha+cosbeta=0=sinalpha+sinbeta,cos2alpha+cos2beta is equal to a) -2sin(alpha+beta) b) 2cos(alpha+beta) c) 2sin(alpha-beta) d) -2cos(alpha+beta)

If alpha, beta in (0, (pi)/(2)), sin alpha = (4)/(5) and cos (alpha + beta) = -(12)/(13) , then sin beta is equal to

If A(alpha, beta) = [(cos alpha, sin alpha, 0),(- sin alpha, cos alpha, 0),(0,0,e^beta)] then A(alpha, beta)^(-1) is equal to a) A(-alpha, -beta) b) A(-alpha,beta) c) A(alpha, - beta) d) A(alpha, beta)

If A=[(cos^2alpha,cosalphasinalpha),(cosalphasinalpha,sin^2alpha)] and B=[(cos^2beta,cosbetasinbeta),(cosbetasinbeta,sin^2beta)] . Show that AB=[(cosalphacosbetacos(alpha-beta),sinbetacosalphacos(alpha-beta)),(cosbetasinalphacos(alpha-beta),sinbetasinalphacos(alpha-beta))]

If A_(alpha)=[(cos alpha,sinalpha),(-sinalpha,cosalpha)] then A_(alpha) A_(beta) is equal to A) A_(alpha beta) B) A_(alpha +beta) C) A_(alpha - beta) D)None of these