Trigonometry Examples

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

Step 1

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

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

Step 2

Separate fractions.

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

Step 3

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

$\frac{1}{\csc (x)} \cdot \frac{\tan (x)}{\frac{1}{\sin (x)}} + \frac{\tan (x)}{\sec^{2} (x)}$

Step 4

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

$\frac{1}{\csc (x)} \cdot \frac{\frac{\sin (x)}{\cos (x)}}{\frac{1}{\sin (x)}} + \frac{\tan (x)}{\sec^{2} (x)}$

Step 5

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

$\frac{1}{\csc (x)} \cdot \frac{\sin (x) \sin (x)}{\cos (x)} + \frac{\tan (x)}{\sec^{2} (x)}$

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

Add $1$ and $1$ .

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

Step 11

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

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

Step 12

Separate fractions.

$\sin^{2} (x) \tan (x) + \frac{1}{\sec (x)} \cdot \frac{\tan (x)}{\sec (x)}$

Step 13

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

$\sin^{2} (x) \tan (x) + \frac{1}{\sec (x)} \cdot \frac{\tan (x)}{\frac{1}{\cos (x)}}$

Step 14

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

$\sin^{2} (x) \tan (x) + \frac{1}{\sec (x)} \cdot \frac{\frac{\sin (x)}{\cos (x)}}{\frac{1}{\cos (x)}}$

Step 15

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

$\sin^{2} (x) \tan (x) + \frac{1}{\sec (x)} (\frac{\sin (x)}{\cos (x)} \cos (x))$

Step 16

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

$\sin^{2} (x) \tan (x) + \frac{1}{\sec (x)} (\frac{\sin (x)}{\cos (x)} \cdot \frac{\cos (x)}{1})$

Step 17

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

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Step 17.1

Cancel the common factor.

$\sin^{2} (x) \tan (x) + \frac{1}{\sec (x)} (\frac{\sin (x)}{\cos (x)} \cdot \frac{\cos (x)}{1})$

Step 17.2

Rewrite the expression.

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

Step 18

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

$\sin^{2} (x) \tan (x) + \frac{1}{\frac{1}{\cos (x)}} \sin (x)$

Step 19

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

$\sin^{2} (x) \tan (x) + 1 \cos (x) \sin (x)$

Step 20

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

$\sin^{2} (x) \tan (x) + \cos (x) \sin (x)$