Simplify tan(theta)^2*csc(theta)^2

$\tan^{2} (θ) \cdot \csc^{2} (θ)$

Step 1

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

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

Step 2

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

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

Step 3

Simplify the expression.

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

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

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

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

One to any power is one.

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

Step 4

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

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

Step 5

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

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{\cos^{2} (θ)}$

Step 6

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

$\frac{1^{2}}{\cos^{2} (θ)}$

Step 7

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

${(\frac{1}{\cos (θ)})}^{2}$

Step 8

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

$\sec^{2} (θ)$