Trigonometry Examples

$\frac{\sec (x) \tan (x)}{\cot (x) \csc (x)}$

Step 1

Separate fractions.

$\frac{\tan (x)}{\csc (x)} \cdot \frac{\sec (x)}{\cot (x)}$

Step 2

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

$\frac{\tan (x)}{\csc (x)} \cdot \frac{\sec (x)}{\frac{\cos (x)}{\sin (x)}}$

Step 3

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

$\frac{\tan (x)}{\csc (x)} \cdot \frac{\frac{1}{\cos (x)}}{\frac{\cos (x)}{\sin (x)}}$

Step 4

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

$\frac{\tan (x)}{\csc (x)} (\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)})$

Step 5

Simplify.

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Step 5.1

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

$\frac{\tan (x)}{\csc (x)} (\sec (x) \frac{\sin (x)}{\cos (x)})$

Step 5.2

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

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

Step 6

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

$\frac{\tan (x)}{\frac{1}{\sin (x)}} (\sec (x) \tan (x))$

Step 7

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

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

Step 8

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

$\frac{\sin (x) \sin (x)}{\cos (x)} (\sec (x) \tan (x))$

Step 9

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

$\sin (x) \tan (x) (\sec (x) \tan (x))$

Step 10

Multiply $\sin (x) \tan (x) (\sec (x) \tan (x))$ .

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Step 10.1

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

$\sin (x) (\tan^{1} (x) \tan (x)) \sec (x)$

Step 10.2

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

$\sin (x) (\tan^{1} (x) \tan^{1} (x)) \sec (x)$

Step 10.3

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

$\sin (x) {\tan (x)}^{1 + 1} \sec (x)$

Step 10.4

Add $1$ and $1$ .

$\sin (x) \tan^{2} (x) \sec (x)$