Answer:
see the explanation
Step-by-step explanation:
we have
[tex]cos(2\alpha)=\frac{1-tan^2(\alpha)}{1+tan^2(\alpha)}[/tex]
Prove
Remember that
[tex]cos(2\alpha)=cos^2(\alpha)-sin^2(\alpha)[/tex]
[tex]tan(\alpha)=\frac{sin(\alpha)}{cos(\alpha)}[/tex]
substitute in the expression above
[tex]cos^2(\alpha)-sin^2(\alpha)=\frac{1-(\frac{sin(\alpha)}{cos(\alpha)})^2}{1+(\frac{sin(\alpha)}{cos(\alpha)})^2}[/tex]
[tex]cos^2(\alpha)-sin^2(\alpha)=\frac{(\frac{cos^2(\alpha)-sin^2(\alpha)}{cos^2(\alpha)}}{(\frac{cos^2(\alpha)+sin^2(\alpha)}{cos^2(\alpha)}}}[/tex]
Remember that
[tex]cos^2(\alpha)+sin^2(\alpha)=1[/tex]
Simplify
[tex]cos^2(\alpha)-sin^2(\alpha)=cos^2(\alpha)-sin^2(\alpha)[/tex] ---> is proved
