Trigonometry Examples

$\cot^{2} (r) \cdot (1 + \tan^{2} (r))$

Step 1

Rearrange terms.

$\cot^{2} (r) \cdot (\tan^{2} (r) + 1)$

Step 2

Apply pythagorean identity.

$\cot^{2} (r) \cdot \sec^{2} (r)$

Step 3

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

${(\frac{\cos (r)}{\sin (r)})}^{2} \cdot \sec^{2} (r)$

Step 4

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

${(\frac{\cos (r)}{\sin (r)})}^{2} \cdot {(\frac{1}{\cos (r)})}^{2}$

Step 5

Simplify the expression.

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Step 5.1

Apply the product rule to $\frac{\cos (r)}{\sin (r)}$ .

$\frac{\cos^{2} (r)}{\sin^{2} (r)} \cdot {(\frac{1}{\cos (r)})}^{2}$

Step 5.2

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

$\frac{\cos^{2} (r)}{\sin^{2} (r)} \cdot \frac{1^{2}}{\cos^{2} (r)}$

Step 5.3

One to any power is one.

$\frac{\cos^{2} (r)}{\sin^{2} (r)} \cdot \frac{1}{\cos^{2} (r)}$

Step 6

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

$\frac{\cos^{2} (r)}{\sin^{2} (r) \cos^{2} (r)}$

Step 7

Cancel the common factor of $\cos^{2} (r)$ .

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Step 7.1

Cancel the common factor.

$\frac{\cos^{2} (r)}{\sin^{2} (r) \cos^{2} (r)}$

Step 7.2

Rewrite the expression.

$\frac{1}{\sin^{2} (r)}$

Step 8

Rewrite $1$ as $1^{2}$ .

$\frac{1^{2}}{\sin^{2} (r)}$

Step 9

Rewrite $\frac{1^{2}}{\sin^{2} (r)}$ as ${(\frac{1}{\sin (r)})}^{2}$ .

${(\frac{1}{\sin (r)})}^{2}$

Step 10

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

$\csc^{2} (r)$