Simplify (tan(theta)^2+1)/(tan(theta)^2) ?

$\frac{\tan^{2} (θ) + 1}{\tan^{2} (θ)}$ ?

Step 1

Apply pythagorean identity.

$\frac{\sec^{2} (θ)}{\tan^{2} (θ)}$

Step 2

Rewrite $\frac{\sec^{2} (θ)}{\tan^{2} (θ)}$ as ${(\frac{\sec (θ)}{\tan (θ)})}^{2}$ .

${(\frac{\sec (θ)}{\tan (θ)})}^{2}$

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

Cancel the common factor.

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

Rewrite the expression.

${(\frac{1}{\sin (θ)})}^{2}$

Step 7

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

$\csc^{2} (θ)$