Simplify (sec(theta)^2)/(tan(theta))-1/(cot(theta))

$\frac{\sec^{2} (θ)}{\tan (θ)} - \frac{1}{\cot (θ)}$

Step 1

Separate fractions.

$\frac{\sec^{2} (θ)}{1} \cdot \frac{1}{\tan (θ)} - \frac{1}{\cot (θ)}$

Step 2

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

$\frac{\sec^{2} (θ)}{1} \cdot \frac{1}{\frac{\sin (θ)}{\cos (θ)}} - \frac{1}{\cot (θ)}$

Step 3

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

$\frac{\sec^{2} (θ)}{1} \cdot \frac{\cos (θ)}{\sin (θ)} - \frac{1}{\cot (θ)}$

Step 4

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

$\frac{\sec^{2} (θ)}{1} \cot (θ) - \frac{1}{\cot (θ)}$

Step 5

Divide $\sec^{2} (θ)$ by $1$ .

$\sec^{2} (θ) \cot (θ) - \frac{1}{\cot (θ)}$

Step 6

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

${(\frac{1}{\cos (θ)})}^{2} \cot (θ) - \frac{1}{\cot (θ)}$

Step 7

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

${(\frac{1}{\cos (θ)})}^{2} \frac{\cos (θ)}{\sin (θ)} - \frac{1}{\cot (θ)}$

Step 8

Apply the product rule to $\frac{1}{\cos (θ)}$ .

$\frac{1^{2}}{\cos^{2} (θ)} \cdot \frac{\cos (θ)}{\sin (θ)} - \frac{1}{\cot (θ)}$

Step 9

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

Factor $\cos (θ)$ out of $\cos^{2} (θ)$ .

$\frac{1^{2}}{\cos (θ) \cos (θ)} \cdot \frac{\cos (θ)}{\sin (θ)} - \frac{1}{\cot (θ)}$

Cancel the common factor.

$\frac{1^{2}}{\cos (θ) \cos (θ)} \cdot \frac{\cos (θ)}{\sin (θ)} - \frac{1}{\cot (θ)}$

Rewrite the expression.

$\frac{1^{2}}{\cos (θ)} \cdot \frac{1}{\sin (θ)} - \frac{1}{\cot (θ)}$

Step 10

Multiply $\frac{1^{2}}{\cos (θ)}$ by $\frac{1}{\sin (θ)}$ .

$\frac{1^{2}}{\cos (θ) \sin (θ)} - \frac{1}{\cot (θ)}$

Step 11

Separate fractions.

$\frac{1^{2}}{\sin (θ)} \cdot \frac{1}{\cos (θ)} - \frac{1}{\cot (θ)}$

Step 12

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

$\frac{1^{2}}{\sin (θ)} \sec (θ) - \frac{1}{\cot (θ)}$

Step 13

Separate fractions.

$\frac{1^{2}}{1} \cdot \frac{1}{\sin (θ)} \sec (θ) - \frac{1}{\cot (θ)}$

Step 14

Convert from $\frac{1}{\sin (θ)}$ to $\csc (θ)$ .

$\frac{1^{2}}{1} \csc (θ) \sec (θ) - \frac{1}{\cot (θ)}$

Step 15

Divide $1^{2}$ by $1$ .

$1^{2} \csc (θ) \sec (θ) - \frac{1}{\cot (θ)}$

Step 16

One to any power is one.

$1 \csc (θ) \sec (θ) - \frac{1}{\cot (θ)}$

Step 17

Multiply $\csc (θ)$ by $1$ .

$\csc (θ) \sec (θ) - \frac{1}{\cot (θ)}$

Step 18

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

$\csc (θ) \sec (θ) - \frac{1}{\frac{\cos (θ)}{\sin (θ)}}$

Step 19

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

$\csc (θ) \sec (θ) - \frac{\sin (θ)}{\cos (θ)}$

Step 20

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

$\csc (θ) \sec (θ) - \tan (θ)$

Simplify (sec(theta)^2)/(tan(theta))-1/(cot(theta))

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