Question

solve this with proper steps ​

Answer 1

Step-by-step explanation:

We have,

$\tt \: \frac{ tan(A) }{1 - cot(A) } + \frac{ cot(A) }{1 - tan(A) }$

$\tt = \: \frac{ tan(A) }{1 - cot(A) } + \frac{ cot(A) }{1 - \dfrac{1}{ cot(A)} }$

$\tt = \: \frac{ tan(A) }{1 - cot(A) } + \frac{ cot(A) }{ \dfrac{ cot(A) - 1}{ cot(A)} }$

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

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

$\tt = \frac{ \dfrac{1}{cot(A)} - cot^{2} (A) }{1 - cot(A) }$

$\tt = \frac{ 1 - cot^{3} (A) }{ cot(A) (1 - cot(A)) }$

$\tt = \frac{ \{1 - cot (A) \} \{ 1 +cot(A) +cot ^{2} (A)\} }{ cot(A) (1 - cot(A)) }$

$\tt = \frac{ 1 +cot(A) +cot ^{2} (A)}{ cot(A) }$

$\tt = tan(A) + 1 +cot(A)$

$\red{ \tt = 1 + tan(A) +cot(A)}$

Now,

$\tt = 1 + \frac{sin(A)}{ cos(A) } + \frac{cos(A)}{ sin(A) }$

$\tt = \frac{ sin(A) cos(A) +cos^{2} (A) +sin^{2} (A)}{ sin(A) cos(A) }$

$\red{ \tt = 1 + cosec(A) sec(A)}$