If `sinalpha sinbeta-cosalphacosbeta+1=0,` then prove that `1+cotalphatanbeta=0`

If cosalpha+cosbeta=0=sinalpha+sinbeta, then prove that cos2alpha+cos2beta=-2cos(alpha+beta) .

If (2sinalpha)/(1+cosalpha+sinalpha)=y , then prove that (1-cosalpha+sinalpha)/(1+sinalpha) is also equal to y.

If costheta=(cosalphacosbeta)/(1-sinalphasinbeta) then prove that tan((theta)/(2))=(tan((alpha)/(2))-tan((beta)/(2)))/(1-tan((alpha)/(2))tan((beta)/(2)))

If A(alpha)=[{:(cosalpha,sinalpha),(-sinalpha,cosalpha):}] then prove that A(alpha)A(-alpha)=I .

Evaluate |{:(cosalphacosbeta,cosalphasinbeta,-sinalpha),(-sinbeta,cosbeta,0),(sinalphacosbeta,sinalphasinbeta,cosalpha):}|

If theta+beta=alphaandsintheta=ksinbeta then prove that tantheta=(ksinalpha)/(1+kcosalpha)andtanbeta=(sinalpha)/(k+cosalpha) .

Evaluate Delta=|{:(0,sinalpha,-cosalpha),(-sinalpha,0,sinbeta),(cosalpha,-sinbeta,0):}|

Let alpha and beta be nonzero real numbers such that 2(cosbeta-cosalpha)+cosalphacosbeta=1 . Then which of the following is/are true?

Using properties of determinants in Exercise 11 to 15 prove that |{:(sinalpha,cosalpha,cos(alpha+delta)),(sinbeta,cosbeta,cos(beta+delta)),(singamma,cosgamma,cos(gamma+delta)):}|=0

If A=[{:(cosalpha,sinalpha),(-sinalpha,cosalpha):}],andA^(-1)=A' , find value of alpha .