Question

prove that tan/1-cot. + cot/1- tan. = 1 +tan+cot​

Answer 1

any value of tan, cot

Answer 2

$lhs \\ \frac{tan}{1 - cot} + \frac{cot}{1 - tan} \\ \frac{ \frac{sin}{cos} }{ \frac{sin - cos} {sin} } + \frac{ \frac{cos}{sin} }{ \frac{cos - sin}{cos} } \\ \frac{ {sin}^{2} }{cos( sin- cos)} + \frac{ {cos}^{2} }{sin(cos - sin)} \\ \frac{ {sin}^{2} }{cos( sin- cos)} - \frac{ {cos}^{2} }{sin(sin - cos)} \\ \frac{ {sin}^{3} \: - {cos}^{3} }{sin \times cos(sin - \: cos)} \\ \frac{(sin \: - cos)( {sin}^{2} + {cos}^{2} + sin \times cos )}{ sin \times cos( sin-cos )} \\ \frac{1 + sin \times cos}{sin \times cos \: } = 1 + cosec \times sec$

$rhs \\ 1 + tan + cot \\ 1 + \frac{sin}{cos} + \frac{cos}{sin} \\ \frac{sin \times cos + {sin}^{2} + {cos}^{2} }{sin \times cos \: } \\ \frac{sin \times cos \: + 1 }{sin \times cos \:} = 1 + cosec \times sec$