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

Jan 3, 2018

$- 1$

#### Explanation:

One of the fundamental identities is $1 + {\cot}^{2} \left(x\right) = {\csc}^{2} \left(x\right)$.

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

Replace ${\csc}^{2} \left(x\right)$ with $1 + {\cot}^{2} \left(x\right)$:

${\cot}^{2} \left(x\right) - \left(1 + {\cot}^{2} \left(x\right)\right)$

=cot^2(x)-1-cot^2(x))

$= - 1$

Jan 3, 2018

${\cot}^{2} x - {\csc}^{2} x = - 1$

#### Explanation:

Use the identity $1 + {\cot}^{2} x = {\csc}^{2} x$

Subtract ${\cot}^{2} x$ to both sides:

$1 = {\csc}^{2} x - {\cot}^{2} x$

Rewrite as:

$1 = - {\cot}^{2} x + {\csc}^{2} x$

Divde both sides by a negative $\left(-\right)$

$- 1 = {\cot}^{2} x - {\csc}^{2} x$

Jan 3, 2018

$- 1$

#### Explanation:

Another way is to reduce all the functions to sine and cosines

${\cot}^{2} x - {\csc}^{2} x = {\cos}^{2} \frac{x}{\sin} ^ 2 x - \frac{1}{\sin} ^ 2 x$

$= \frac{{\cos}^{2} x - 1}{\sin} ^ 2 x$

$= - \frac{1 - {\cos}^{2} x}{\sin} ^ 2 x = - {\sin}^{2} \frac{x}{\sin} ^ 2 x$

$= - 1$