[tex]\tan5u\tan3u=\dfrac{\tan^24u\tan^2u}{1-\tan^24u\tan^2u}[/tex]
This is not true (assuming I've understood the question correctly).
Consider [tex]u=\dfrac\pi4[/tex]. Then
[tex]\tan\dfrac{5\pi}4\tan\dfrac{3\pi}4=1(-1)=-1[/tex]
while
[tex]\dfrac{\tan^2\pi\tan^2\frac\pi4}{1-\tan^2\pi\tan^2\frac\pi4}=\dfrac{0(1)}{1-0(1)}=0[/tex]
