Trigonometry Examples

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

Step 1

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

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

Step 2

Separate fractions.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Multiply $\cos (x) (\cos (x) \cot (x))$ .

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

Add $1$ and $1$ .

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

Step 11

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

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

Step 12

Separate fractions.

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

Step 13

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

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

Step 14

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

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

Step 15

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

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

Step 16

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

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

Step 17

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

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Step 17.1

Cancel the common factor.

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

Step 17.2

Rewrite the expression.

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

Step 18

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

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

Step 19

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

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

Step 20

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

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