# How do you simplify tan^2 x- cot^2x?

Nov 6, 2015

${\tan}^{2} x - {\cot}^{2} x = \frac{{\sin}^{2} x}{{\cos}^{2} x} - \frac{{\cos}^{2} x}{{\sin}^{2} x}$

$= \frac{{\sin}^{4} x - {\cos}^{4} x}{{\sin}^{2} x \cdot {\cos}^{2} x}$

=((sin^2x+cos^2x)(sin^2x-cos^2x))/((1-cos^2x)(1-sin^2x)

=((1)(sinx+cosx)(sinx-cosx))/((1+cosx)(1-cosx)(1+sinx)(1-sinx)

Does not appear to be able to simplify any further.

Mar 26, 2017

${\tan}^{2} x - {\cot}^{2} x = - 4 \csc \left(2 x\right) \cot \left(2 x\right)$

#### Explanation:

${\tan}^{2} x - {\cot}^{2} x$

$= {\sin}^{2} \frac{x}{\cos} ^ 2 \left(x\right) - {\cos}^{2} \frac{x}{\sin} ^ 2 \left(x\right)$

$= \frac{{\sin}^{4} \left(x\right) - {\cos}^{4} \left(x\right)}{{\cos}^{2} \left(x\right) {\sin}^{2} \left(x\right)}$

$= \frac{\left({\sin}^{2} x + {\cos}^{2} x\right) \left({\sin}^{2} x - {\cos}^{2} x\right)}{\cos \left(x\right) \sin \left(x\right)} ^ 2$

$= \frac{\left(1\right) \left(- \cos \left(2 x\right)\right)}{\frac{1}{4} {\sin}^{2} \left(2 x\right)}$

$= - 4 \cos \frac{2 x}{\sin} ^ 2 \left(2 x\right)$

$= - 4 \csc \left(2 x\right) \cot \left(2 x\right)$

$\left({\sec}^{2} x - {\csc}^{2} x\right)$
$f \left(x\right) = \left({\tan}^{2} x + 1\right) - \left({\cot}^{2} x + 1\right) =$
$f \left(x\right) = {\sec}^{2} x - {\csc}^{2} x$