Calculus Examples

$\frac{- \cot (t) \csc (t)}{\cos (t)}$

Step 1

Separate fractions.

$\frac{- 1 \csc (t)}{1} \cdot \frac{\cot (t)}{\cos (t)}$

Step 2

Rewrite $\cot (t)$ in terms of sines and cosines.

$\frac{- 1 \csc (t)}{1} \cdot \frac{\frac{\cos (t)}{\sin (t)}}{\cos (t)}$

Step 3

Rewrite $\frac{\frac{\cos (t)}{\sin (t)}}{\cos (t)}$ as a product.

$\frac{- 1 \csc (t)}{1} (\frac{\cos (t)}{\sin (t)} \cdot \frac{1}{\cos (t)})$

Step 4

Cancel the common factor of $\cos (t)$ .

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Step 4.1

Cancel the common factor.

$\frac{- 1 \csc (t)}{1} (\frac{\cos (t)}{\sin (t)} \cdot \frac{1}{\cos (t)})$

Step 4.2

Rewrite the expression.

$\frac{- 1 \csc (t)}{1} \cdot \frac{1}{\sin (t)}$

Step 5

Simplify the expression.

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Step 5.1

Divide $- 1 \csc (t)$ by $1$ .

$- 1 \csc (t) \frac{1}{\sin (t)}$

Step 5.2

Rewrite $- 1 \csc (t)$ as $- \csc (t)$ .

$- \csc (t) \frac{1}{\sin (t)}$

Step 6

Rewrite $\csc (t)$ in terms of sines and cosines.

$- \frac{1}{\sin (t)} \cdot \frac{1}{\sin (t)}$

Step 7

Multiply $- \frac{1}{\sin (t)} \cdot \frac{1}{\sin (t)}$ .

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Step 7.1

Multiply $\frac{1}{\sin (t)}$ by $\frac{1}{\sin (t)}$ .

$- \frac{1}{\sin (t) \sin (t)}$

Step 7.2

Raise $\sin (t)$ to the power of $1$ .

$- \frac{1}{\sin^{1} (t) \sin (t)}$

Step 7.3

Raise $\sin (t)$ to the power of $1$ .

$- \frac{1}{\sin^{1} (t) \sin^{1} (t)}$

Step 7.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{1}{{\sin (t)}^{1 + 1}}$

Step 7.5

Add $1$ and $1$ .

$- \frac{1}{\sin^{2} (t)}$

Step 8

Rewrite $1$ as $1^{2}$ .

$- \frac{1^{2}}{\sin^{2} (t)}$

Step 9

Rewrite $\frac{1^{2}}{\sin^{2} (t)}$ as ${(\frac{1}{\sin (t)})}^{2}$ .

$- {(\frac{1}{\sin (t)})}^{2}$

Step 10

Convert from $\frac{1}{\sin (t)}$ to $\csc (t)$ .

$- \csc^{2} (t)$