​$$\text{The value of } 3 + \tan^2 \phi + \cot^2 \phi - \sec^2 \phi \cosec^2 \phi \text{ is:}$$​

Correct option is A

Given:

Expression: 3 +$$\tan^2 \phi + \cot^2 \phi - \sec^2 \phi \cosec^2 \phi$$​​

Formula Used:

$$\sec^2 \phi = 1 + \tan^2 \phi \\ \ \\\cosec^2 \phi = 1 + \cot^2 \phi$$​

Solution:

​$$3 + \tan^2 \phi + \cot^2 \phi - (1 + \tan^2 \phi)(1 + \cot^2 \phi)$$​​

​$$= 3 + \tan^2 \phi + \cot^2 \phi - (1 + \tan^2 \phi + \cot^2 \phi + \tan^2 \phi \cot^2 \phi)$$​​

=$$3 - 1 - \tan^2 \phi \cot^2 \phi$$​​

Since $$\tan \phi \cot \phi = 1, \tan^2 \phi \cot^2 \phi = 1$$​, so the expression becomes:

= 2 - 1 = 1