Assertion (A): $\Big(\dfrac{1 + \tan \theta}{1 + \cot \theta} \Big)^2 = \tan^2 \theta$

Reason (R): tan² θ + sec² θ = 1

1. A is true, R is false

2. A is false, R is true

3. Both A and R are true

4. Both A and R are false

$\Rightarrow \Big(\dfrac{1 + \tan \theta}{1 + \cot \theta} \Big)^2 \\[1em] \Rightarrow \Big(\dfrac{1 + \tan \theta}{1 + \dfrac{1}{\tan \theta}} \Big)^2 \\[1em] \Rightarrow \Big(\dfrac{1 + \tan \theta}{ \dfrac{\tan \theta + 1}{\tan \theta}} \Big)^2 \\[1em] \Rightarrow \tan^2 \theta$

Therefore, assertion (A) is true.

tan² θ + sec² θ = 1 is incorrect

The correct identity is sec² θ - tan² θ = 1

Therefore, reason (R) is false.

A is true, R is false.

Hence, option 1 is the correct option.