Simplify (csc(t)^2)/(csc(t)^2-1)

$\frac{\csc^{2} (t)}{\csc^{2} (t) - 1}$

Step 1

Apply pythagorean identity.

$\frac{\csc^{2} (t)}{\cot^{2} (t)}$

Step 2

Rewrite $\frac{\csc^{2} (t)}{\cot^{2} (t)}$ as ${(\frac{\csc (t)}{\cot (t)})}^{2}$ .

${(\frac{\csc (t)}{\cot (t)})}^{2}$

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Cancel the common factor of $\sin (t)$ .

Cancel the common factor.

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

Rewrite the expression.

${(\frac{1}{\cos (t)})}^{2}$

Step 7

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

$\sec^{2} (t)$