Resolução:
[tex3]\bullet tgx=t\rightarrow \frac{senx}{cosx}=t\rightarrow senx=t.cosx[/tex3]
[tex3]\bullet sen^{2}x+cos^{2}x=1[/tex3]
[tex3](t.cosx)^{2}+cos^{2}x=1[/tex3]
[tex3]t^{2}.cos^{2}x+cos^{2}x=1[/tex3]
[tex3]cos^{2}x(t^{2}+1)=1[/tex3]
[tex3]cos^{2}x=\frac{1}{t^{2}+1}[/tex3]
[tex3]\bullet M=sen2x+cos2x[/tex3]
[tex3]M=(2senx.cosx)+(2cos^{2}x-1)[/tex3]
[tex3]M=2(tcosx).cosx+2cos^{2}x-1[/tex3]
[tex3]M=2tcos^{2}x+2cos^{2}x-1[/tex3]
[tex3]M=\frac{2t}{t^{2}+1}+\frac{2}{t^{2}+1}-1[/tex3]
[tex3]M=\frac{2t+2}{t^{2}+1}-1[/tex3]
[tex3]M=\frac{2t+2-t^{2}-1}{t^{2}+1}[/tex3]
[tex3]\boxed{M=\frac{1+2t-t^{2}}{1+t^{2}}}[/tex3]