Prove that: `(cosalpha+cosbeta)^2+(sinalpha+sinbeta)^2=4cos^2((alpha-beta)/2)`

The determinant : |(cos(alpha+beta),-sin(alpha+beta),cos2beta),(sinalpha,cosalpha,sinbeta),(-cosalpha,sinalpha,cosbeta)|=0 is independent of :

Evaluate {:|( cos alpha cos beta , cos alpha sin beta , -sin alpha ),( -sin beta , cos beta, 0),( sin alpha cos beta, sin alpha sin beta, cos alpha ) |:} =1

Prove that sin^(4) alpha + cos^(4) alpha + 2 sin^(2) alpha cos^(2) alpha = 1 .

Let alpha,beta be such that piltalpha-betalt3pi . If sinalpha+sinbeta=-(21)/(65)andcosalpha+cosbeta=-(17)/(65) , then the value of cos.(alpha-beta)/(2) is :

Let alpha, beta be such that pi lt=alpha lt=beta lt=3pi if sinalpha + sinbeta = -21/65 and cosalpha + cosbeta = -27/65 , then the value of cos""(alpha - beta)/65 is :

IF {:|(sinalpha,cosbeta),(cos alpha,sin beta)|=1/2 , where alpha and beta are acute angles then write the value of alpha + beta.

If A=[[cos alpha, -sin alpha] , [sin alpha, cos alpha]], B=[[cos2beta, sin 2beta] , [sin 2 beta, -cos2beta]] where 0 lt beta lt pi/2 then prove that BAB=A^(-1) Also find the least positive value of alpha for which BA^4B= A^(-1)