Evaluate cot(theta)^2(1+tan(theta)^2)

$\cot^{2} (θ) (1 + \tan^{2} (θ))$

Step 1

Rearrange terms.

$\cot^{2} (θ) (\tan^{2} (θ) + 1)$

Step 2

Apply pythagorean identity.

$\cot^{2} (θ) \sec^{2} (θ)$

Step 3

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

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

Step 4

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

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

Step 5

Simplify the expression.

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

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

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

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

One to any power is one.

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

Step 6

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

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

Step 7

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{\sin^{2} (θ)}$

Step 8

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

$\frac{1^{2}}{\sin^{2} (θ)}$

Step 9

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

${(\frac{1}{\sin (θ)})}^{2}$

Step 10

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

$\csc^{2} (θ)$