Question

Use Pythagorean identities to rewrite each of the following trigonometric expressions.

${\cot }^{2}x=\_\_\_\_\_$

Short answer

The result is$\frac{1}{{\sin }^{2}x}-1$.

Step-by-step solution

Step 1. Given information.

Consider the given expression.

${\cot }^{2}x$

Step 2. Apply the Pythagorean identities to rewrite the expression.`

The Pythagorean identity is defined as,

${\sin }^{2}x+{\cos }^{2}x=1$

Use the identity to write the given expression.

$\begin{array}{rcl}{\sin }^{2}x+{\cos }^{2}x& =& 1\\ {\cos }^{2}x& =& 1-{\sin }^{2}x\\ \frac{{\cos }^{2}x}{{\sin }^{2}x}& =& \frac{1-{\sin }^{2}x}{{\sin }^{2}x}\\ {\cot }^{2}x& =& \frac{1}{{\sin }^{2}x}-1\end{array}$