${(1 - t a n A)}^{2} + {(1 + t a n A)}^{2} = 2 s e c^{2} A$

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Solution

$(1 - tan A)^{2} + (1 + tan A)^{2}$
$\Rightarrow 1 + {tan}^{2} A - 2 tan A + 1 + {tan}^{2} A + 2 tan A$
$\Rightarrow 2 (1 + {tan}^{2} A)$
$\Rightarrow 2 {sec}^{2} A$ .

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