Simplify (tan(x))/(sec(x)^2)+(tan(x))/(csc(x)^2)

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

Step 1

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

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

Step 2

Separate fractions.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Separate fractions.

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

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

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

Step 17

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

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

Step 18

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

$\cos (x) \sin (x) + 1 \sin (x) (\sin (x) \tan (x))$

Step 19

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

$\cos (x) \sin (x) + \sin (x) (\sin (x) \tan (x))$

Step 20

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify (tan(x))/(sec(x)^2)+(tan(x))/(csc(x)^2)

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