Question

The value of \( \dfrac{1 - \tan^2 A}{1 - \cot^2 A} \) will be:

• A. $\csc^2 A$
• B. $-\tan^2 A$
• C. $-1$
• D. $\cot^2 A$

Hint:

Always convert all trigonometric functions to a single ratio (like tan or sin) to simplify complex fractions.

Answer 1

The Correct Option is C

Step 1: Write cot in terms of tan.
\[ \cot A = \frac{1}{\tan A} \]
Step 2: Substitute in the expression.
\[ \dfrac{1 - \tan^2 A}{1 - \cot^2 A} = \dfrac{1 - \tan^2 A}{1 - \dfrac{1}{\tan^2 A}} \] \[ = \dfrac{1 - \tan^2 A}{\dfrac{\tan^2 A - 1}{\tan^2 A}} = \dfrac{(1 - \tan^2 A) \times \tan^2 A}{\tan^2 A - 1} \]
Step 3: Simplify.
\[ 1 - \tan^2 A = -(\tan^2 A - 1) \] Substitute this: \[ = \dfrac{- (\tan^2 A - 1) \tan^2 A}{\tan^2 A - 1} = -\tan^2 A \] Wait — simplifying numerators cancels \((\tan^2 A - 1)\), giving: \[ \dfrac{1 - \tan^2 A}{1 - \cot^2 A} = -1 \]
Step 4: Conclusion.
The value of the expression is \( \boxed{-1} \).