$\frac{(1 + \tan^{2} A)}{(1 + \cot^{2} A)} = ?$
(a) tan2 A
(b) cot2 A
(c) sec2 A
(d) cosec2 A

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Solution

(a) tan² A
$\frac{1 + \tan^{2} A}{1 + \cot^{2} A} = \frac{1 + \frac{{sin}^{2} A}{{cos}^{2} A}}{1 + \frac{{cos}^{2} A}{{sin}^{2} A}} = \frac{\frac{{cos}^{2} A + s {in}^{2} A}{{cos}^{2} A}}{\frac{{sin}^{2} A + c {os}^{2} A}{{sin}^{2} A}} = \frac{\frac{1}{{cos}^{2} A}}{\frac{1}{{sin}^{2} A}} (∵ c {os}^{2} A + s {in}^{2} A = 1) = \frac{{sin}^{2} A}{{cos}^{2} A} = {tan}^{2} A$

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