Find the value of

$\frac{1-{\cot \theta }^2}{{\tan \theta }^2-1}.$

The correct answer is cot²θ.

Step 1: Analyze the given expression.

$\frac{1-{\cot \theta }^2}{{\tan \theta }^2-1}$

Step 2: Express cot θ in terms of tan θ.
Using the reciprocal trigonometric identity, we know that:
$\cot \theta =\frac{1}{\tan \theta }$
Squaring both sides gives:
${\cot \theta }^2=\frac{1}{{\tan \theta }^2}$

Step 3: Simplify the numerator.

$1-{\cot \theta }^2=1-\frac{1}{{\tan \theta }^2}=\frac{{\tan \theta }^2-1}{{\tan \theta }^2}$

Step 4: Substitute the simplified numerator into the original expression.

$\frac{1-{\cot \theta }^2}{{\tan \theta }^2-1}=\frac{\left(\frac{{\tan \theta }^2-1}{{\tan \theta }^2}\right)}{{\tan \theta }^2-1}$

Step 5: Cancel common terms.
Canceling out the non-zero term (tan²θ - 1) present in both the numerator and denominator:

$=\frac{1}{{\tan \theta }^2}={\cot \theta }^2$

Thus, the final simplified value of the given trigonometric expression is cot²θ.