If `cos(alpha+beta)=4/5`,`sin(alpha-beta)=5/13 and if `0 < `alpha` < `pi`/4`,`0 < `beta` < `pi`/4`, `find the value of `tan2alpha`.

Prove that (cos alpha-cos beta)^2+(sin alpha-sin beta)^2=4sin^2((alpha-beta)/2)

if xcosalpha+ysinalpha=k=xcosbeta+ysinbeta ,show that x/(cos((alpha+beta)/2))=y/(sin((alpha+beta)/2))=k/(cos((alpha-beta)/2))

If (alpha_1,beta_1),(alpha_2,beta_2),(alpha_3,beta_3)and(alpha_4,beta_4) are the vertices of a parallelogram,show that alpha_1+alpha_3=alpha_2+alpha_4 and beta_1+beta_3=beta_2+beta_4 .

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

The angles of depression of two consecutive kilometer posts on a straight road as seen from an aeroplane above it are alpha and beta(alpha>beta) .Find the height of the aeroplane above the road.

if tanalpha , tanbeta are the roots of x^2+px+q=0 ,show that sin^2(alpha+beta)+p sin(alpha+beta)cos(alpha+beta)+q cos^2(alpha+beta)=q

If cos alpha=13/14 and cos beta=1/7 where alpha and beta are acute angles,show that alpha-beta=pi/3 .

If A=[(cos alpha, -sin alpha),(sin alpha, cos alpha)] and A+A^(T) =l , write down the general values of alpha .

If cos(alpha-beta)+cos(beta-gamma)+cos(gamma-alpha)=-3/2 ,show that cosalpha+cosbeta+cosgamma=sinalpha+sinbeta+singamma=0