Use the diagram to prove the double angle formula, where $t=\tan \theta$: ${\displaystyle \tan 2\theta =\frac{2t}{1-t^2},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\sin 2\theta =\frac{2t}{1+t^2},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\cos 2\theta =\frac{1-t^2}{1+t^2}}$