Answer
$\tan^{2}x-\cot^{2}x=\sec^{2}x-\csc^{2}x$
Work Step by Step
$\tan^{2}x-\cot^{2}x=\sec^{2}x-\csc^{2}x$
On the left side of the equation, substitute $\tan^{2}x$ with $\sec^{2}x-1$ and $\cot^{2}x$ with $\csc^{2}x-1$:
$\sec^{2}x-1-(\csc^{2}x-1)=\sec^{2}x-\csc^{2}x$
Simplify and the identity is proved:
$\sec^{2}x-1-\csc^{2}x+1=\sec^{2}x-\csc^{2}x$
$\sec^{2}x-\csc^{2}x=\sec^{2}x-\csc^{2}x$
