we have the equation
[tex](cotx+tanx)^2=csc^2x+sec^2x[/tex]
Remember that
[tex]\begin{gathered} tan^2x+1=sec^2x \\ cot^2x+1=csc^2x \end{gathered}[/tex][tex]\begin{gathered} tanx=\frac{sinx}{cosx} \\ \\ cotx=\frac{cosx}{sinx} \end{gathered}[/tex]
substitute
[tex](cotx+tanx)^2=(\frac{cosx}{sinx}+\frac{sinx}{cosx})^2=(\frac{cos^2x+sin^2x}{sinxcosx})^2=(\frac{1}{sinxcosx})^2=(cscxsecx)^2[/tex]
and
[tex](cscxsecx)^2=csc^2x*sec^2x[/tex]
substitute the identity
[tex]csc^2x*sec^2x=(cot^2x+1)(tan^2x+1)=cot^2xtan^2x+cot^2x+tan^2x+1)=sec^2x+csc^2x[/tex]
Remember that
[tex]cot^2xtan^2x=1[/tex]
therefore
[tex]sec^2x+csc^2x=sec^2x+csc^2x\text{ -----> is proved}[/tex]
