Trigonometry Examples

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

Step 1

Simplify with factoring out.

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Step 1.1

Reorder $1$ and $- \csc^{2} (x)$ .

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

Step 1.2

Factor $- 1$ out of $- \csc^{2} (x)$ .

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

Step 1.3

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{\tan (\frac{π}{2} - x) \sec (x)}{- \csc^{2} (x) - 1 \cdot - 1}$

Step 1.4

Factor $- 1$ out of $- \csc^{2} (x) - 1 \cdot - 1$ .

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

Step 2

Apply pythagorean identity.

$\frac{\tan (\frac{π}{2} - x) \sec (x)}{- \cot^{2} (x)}$

Step 3

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

$\frac{\tan (\frac{π}{2} - x) \sec (x)}{- (\cot (x) \cot (x))}$

Step 4

Separate fractions.

$\frac{\sec (x)}{- (\cot (x))} \cdot \frac{\tan (\frac{π}{2} - x)}{\cot (x)}$

Step 5

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

$\frac{\sec (x)}{- (\cot (x))} \cdot \frac{\tan (\frac{π}{2} - x)}{\frac{\cos (x)}{\sin (x)}}$

Step 6

Rewrite $\tan (\frac{π}{2} - x)$ in terms of sines and cosines.

$\frac{\sec (x)}{- (\cot (x))} \cdot \frac{\frac{\sin (\frac{π}{2} - x)}{\cos (\frac{π}{2} - x)}}{\frac{\cos (x)}{\sin (x)}}$

Step 7

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

$\frac{\sec (x)}{- (\cot (x))} (\frac{\sin (\frac{π}{2} - x)}{\cos (\frac{π}{2} - x)} \cdot \frac{\sin (x)}{\cos (x)})$

Step 8

Simplify.

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Step 8.1

Convert from $\frac{\sin (\frac{π}{2} - x)}{\cos (\frac{π}{2} - x)}$ to $\tan (\frac{π}{2} - x)$ .

$\frac{\sec (x)}{- (\cot (x))} (\tan (\frac{π}{2} - x) \frac{\sin (x)}{\cos (x)})$

Step 8.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{\sec (x)}{- (\cot (x))} (\tan (\frac{π}{2} - x) \tan (x))$

Step 9

Separate fractions.

$\frac{1}{- 1} \cdot \frac{\sec (x)}{\cot (x)} (\tan (\frac{π}{2} - x) \tan (x))$

Step 10

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

$\frac{1}{- 1} \cdot \frac{\sec (x)}{\frac{\cos (x)}{\sin (x)}} (\tan (\frac{π}{2} - x) \tan (x))$

Step 11

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

$\frac{1}{- 1} \cdot \frac{\frac{1}{\cos (x)}}{\frac{\cos (x)}{\sin (x)}} (\tan (\frac{π}{2} - x) \tan (x))$

Step 12

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

$\frac{1}{- 1} (\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}) (\tan (\frac{π}{2} - x) \tan (x))$

Step 13

Simplify.

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Step 13.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{1}{- 1} (\sec (x) \frac{\sin (x)}{\cos (x)}) (\tan (\frac{π}{2} - x) \tan (x))$

Step 13.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{1}{- 1} (\sec (x) \tan (x)) (\tan (\frac{π}{2} - x) \tan (x))$

Step 14

Divide $1$ by $- 1$ .

$- (\sec (x) \tan (x)) (\tan (\frac{π}{2} - x) \tan (x))$

Step 15

Multiply $- \sec (x) \tan (x) (\tan (\frac{π}{2} - x) \tan (x))$ .

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Step 15.1

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

$- \sec (x) (\tan^{1} (x) \tan (x)) \tan (\frac{π}{2} - x)$

Step 15.2

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

$- \sec (x) (\tan^{1} (x) \tan^{1} (x)) \tan (\frac{π}{2} - x)$

Step 15.3

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

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

Step 15.4

Add $1$ and $1$ .

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