#sec(theta+alpha)+sec(theta-alpha)=2sec(theta)# can be written as
#1/cos(theta+alpha)+1/cos(theta-alpha)=2/costheta#
or #(cos(theta+alpha)+cos(theta-alpha))/(cos(theta+alpha)cos(theta-alpha))=2/cos(theta)#
or#(2xx2costhetacosalpha)/(cos(theta+alpha+theta-alpha)+cos(theta+alpha-theta+alpha))=2/costheta#
or #(2costhetacosalpha)/(cos2theta+cos2alpha)=1/costheta#
or #2cos^2thetacosalpha=cos2theta+cos2alpha#
or #2cos^2thetacosalpha=2cos^2theta-1+2cos^2alpha-1#
or #2cos^2theta(cosalpha-1)=2cos^2alpha-2#
or #cos^2theta=(cos^2alpha-1)/(cosalpha-1)=cosalpha+1#
or #cos^2theta=2cos^2(alpha/2)#
and #costheta=sqrt2cos(alpha/2)#