Which of the following options is/are equal to tan2A+cot2A+2 ?

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Which of the following options is/are equal to $t a n^{2} A + c o t^{2} A + 2$ ?

Options:

A .

\frac{1}{c o s^{2} A s i n^{2} A}

B .

s e c^{2} A c o s e c^{2} A

C .

t a n^{2} A c o t^{2} A

D .

s e c^{2} A + c o s e c^{2} A

Answer: Option A
:
A, B, and D

$t a n^{2} A + c o t^{2} A + 2 can be written as (1 + t a n^{2} A) + (1 + c o t^{2} A)$
We know that, $1 + t a n^{2} A = s e c^{2} A and 1 + c o t^{2} A = c o s e c^{2} A$
$\Rightarrow t a n^{2} A + c o t^{2} A + 2 = s e c^{2} A + c o s e c^{2} A$

$= \frac{1}{c o s^{2} A} + \frac{1}{s i n^{2} A}$

$= \frac{s i n^{2} A + c o s^{2} A}{c o s^{2} A s i n^{2} A} . . . (s i n^{2} A + c o s^{2} A = 1)$

$= \frac{1}{c o s^{2} A s i n^{2} A}$

$= s e c^{2} A c o s e c^{2} A$

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