Trigonometry Examples

$\frac{\csc (- x)}{1 + \tan^{2} (x)}$

Step 1

Rearrange terms.

$\frac{\csc (- x)}{\tan^{2} (x) + 1}$

Step 2

Apply pythagorean identity.

$\frac{\csc (- x)}{\sec^{2} (x)}$

Step 3

Factor $\sec (x)$ out of $\sec^{2} (x)$ .

$\frac{\csc (- x)}{\sec (x) \sec (x)}$

Step 4

Separate fractions.

$\frac{1}{\sec (x)} \cdot \frac{\csc (- x)}{\sec (x)}$

Step 5

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\sec (x)} \cdot \frac{\csc (- x)}{\frac{1}{\cos (x)}}$

Step 6

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

$\frac{1}{\sec (x)} \cdot \frac{\frac{1}{\sin (- x)}}{\frac{1}{\cos (x)}}$

Step 7

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

$\frac{1}{\sec (x)} (\frac{1}{\sin (- x)} \cos (x))$

Step 8

Write $\cos (x)$ as a fraction with denominator $1$ .

$\frac{1}{\sec (x)} (\frac{1}{\sin (- x)} \cdot \frac{\cos (x)}{1})$

Step 9

Simplify.

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Step 9.1

Divide $\cos (x)$ by $1$ .

$\frac{1}{\sec (x)} (\frac{1}{\sin (- x)} \cos (x))$

Step 9.2

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

$\frac{1}{\sec (x)} (\csc (- x) \cos (x))$

Step 10

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\frac{1}{\cos (x)}} (\csc (- x) \cos (x))$

Step 11

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

$1 \cos (x) (\csc (- x) \cos (x))$

Step 12

Multiply $\cos (x)$ by $1$ .

$\cos (x) (\csc (- x) \cos (x))$

Step 13

Since $\csc (- x)$ is an odd function, rewrite $\csc (- x)$ as $- \csc (x)$ .

$\cos (x) (- \csc (x) \cos (x))$

Step 14

Rewrite using the commutative property of multiplication.

$- \cos (x) \csc (x) \cos (x)$

Step 15

Multiply $- \cos (x) \csc (x) \cos (x)$ .

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Step 15.1

Raise $\cos (x)$ to the power of $1$ .

$- (\cos^{1} (x) \cos (x)) \csc (x)$

Step 15.2

Raise $\cos (x)$ to the power of $1$ .

$- (\cos^{1} (x) \cos^{1} (x)) \csc (x)$

Step 15.3

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

$- {\cos (x)}^{1 + 1} \csc (x)$

Step 15.4

Add $1$ and $1$ .

$- \cos^{2} (x) \csc (x)$