Question

prove:tan^2 + cot^2 + 2 = sec^2 . cosec^2

Answer 1

Step-by-step explanation:

LHS..

$tan {}^{2} x + cot {}^{2} x + 2 \\ \\ = \frac{ \sin {}^{2} (x) }{ \cos{}^{2}( x) } + \frac{ \cos {}^{2} (x) }{ \sin {}^{2} (x) } + 2 \\ \\ = \frac{ \sin {}^{4} (x) + \cos {}^{4} (x) + 2 \sin {}^{2} (x) \cos {}^{2} (x) }{ \cos {}^{2} (x) \sin {}^{2} (x) } \\ \\ = \frac{( \sin {}^{2} (x) + \cos {}^{2} (x) ){}^{2} }{{ \cos {}^{2} (x) \sin {}^{2} (x) } } \\ \\ = \frac{1 {}^{2} }{{ \cos {}^{2} (x) \sin {}^{2} (x) } } \\ \\ = \sec {}^{2} (x) \cosec {}^{2} (x)$