The double angle formulae are
{jatex options:inline}cos2x=cos^2x-sin^2x{/jatex}
{jatex options:inline}cos2x=2cos^2x-1{/jatex}
{jatex options:inline}cos2x=1-2sin^2x{/jatex}
{jatex options:inline}sin2x=2sinxcosx{/jatex}
{jatex options:inline}tanx=\frac{2tanx}{1-tan^2x}{/jatex}
The first can be dertived from the cpmound angle formula  {jatex options:inline}cos(A+B)=cosAcosB-sinAsinB{/jatex}.
Pute  {jatex options:inline}A=B=x{/jatex}  then we  {jatex options:inline}cos2x=cos^2x-sin^2x{/jatex}.
To get the second and third equations use  {jatex options:inline}1=cos^2x+sin^2x sin^2x=1-cos^2x{/jatex}.
{jatex options:inline}cos2x=cos^2x-sin^2x=cos^2x-(1-cos^2x)=2cos^2x-1{/jatex}
To obtain the tird use  {jatex options:inline}1=cos^2x+sin^2x cos^2x=1-sin^2x{/jatex}.
{jatex options:inline}cos2x=2cos^2x-1=2(1-sin^2x-1)-1=1-2sin^2x{/jatex}
To obtain the fourth use the compound angle formual  {jatex options:inline}sin(A+B)=sinAcosB+cosAsinB{/jatex}. Put  {jatex options:inline}A=B=x{/jatex}  to obtain
{jatex options:inline}sin2x=sinxcosx+cosxsinx=2sinxcosx{/jatex}.
To obtain thr fifthuse  {jatex options:inline}sin2x=2sinxcosx, \: cos2x=cos^2-sin^2x{/jatex}.
{jatex options:inline}tan2x=\frac{sin2x}{cos2x}=\frac{2sinxcosx}{cos^2x-sin^2x}=\frac{2sinxcosx/cos^2x}{(cos^2x-sin^2x)/cos^2x}=\frac{2tanx}{1-tan^2x}{/jatex}