$(sina+cosa)^2= m^2$
$\Leftrightarrow sin^2a+cos^2a+ 2sina.cosa = m^2$
$\Leftrightarrow sin2a= 2sina.cosa= m^2-1$
$cos2a= \pm\sqrt{1-sin^22a}= \pm \sqrt{1-m^4+2m^2-1}= \pm \sqrt{-m^4+2m^2}$
$tan2a=\frac{sin2a}{cos2a}=\frac{m^2-1}{\pm\sqrt{-m^4+2m^2}}$
$cot2a=\frac{1}{tan2a}=\frac{\pm\sqrt{-m^4+2m^2}}{m^2-1}$