Question

(1+tan²a/1+cot²A) =(1-tan A/1-cot A)²= tan²A

Answer 1

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Answer 2

$i ) \red{ \frac{1+tan^{2} A }{1+cot^{2} A } } \\= \frac{ sec^{2} A }{cosec^{2} A }$

$\underline { \blue { By \: Trigonometric\: Identities : }}$

$\pink { 1) 1+tan^{2} A = sec^{2} A ,}\\\pink{2) 1+cot^{2} A = cosec^{2} A }$

$= \frac{\frac{1}{cos^{2} A}}{\frac{1}{sin^{2} A }}\\= \frac{sin^{2} A }{cos^{2} A } \\= tan ^{2} A \: --(1)$

$ii ) \red { \Big( \frac{1-tanA}{1-cotA}\Big)^{2}}$

$= \Big(\frac{(1-tan A)}{ 1 - \frac{1}{tan A }}\Big)^{2}$

$= \Big( \frac{(1-tan A)}{ \frac{tan A - 1}{tan A }}\Big)^{2}$

$= \Big( \frac{1-tan A}{ \frac{-(1-tan A )}{tan A }}\Big)^{2}$

$= \Big( \frac{1}{\frac{-1}{tan A}}\Big)^{2}\\= tan^{2} A \: ---(2)$

$From\: (1) \:and \:(2) ,$

$\red{ \frac{1+tan^{2} A }{1+cot^{2} A } } \red { = \Big( \frac{1-tanA}{1-cotA}\Big)^{2}} \green {= tan^{2} A }$

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