Question

Prove :
1/1+sin^2 theta +1/1 + cos^2 theta + 1/1+ sec^2 theta + 1/1+ cosec^2 theta = 2​

Answer 1

$lhs = \frac{1}{1 + {sin}^{2} } + \frac{1}{1 + {cos}^{2} } \: + \frac{1}{ {sec}^{2} } + \frac{1}{ {cosec}^{2} } \\ \\ \frac{1}{1 + {sin}^{2} } + \frac{1}{ {cosec }^{2} } + \frac{1}{1 + \frac{1}{ {cos}^{2} } } + \frac{1}{1 + \frac{1}{ {sin}^{2} } } \\ \\ \frac{1}{1 + {sin}^{2} } + \frac{1}{ 1 + {cos}^{2} } + \frac{ {cos}^{2} }{ {cos}^{2} + 1 } + \frac{ {sin}^{2} }{ {sin}^{2} + 1 } \\ \\ \frac{1}{1 + {sin}^{2} } + \frac{ {sin}^{2} }{1 + {sin}^{2} } + \frac{1}{1 + {cos}^{2} } + \frac{ {cos}^{2} }{1 + {cos}^{2} } \\ \\ \frac{1 + {sin}^{2} }{1 + {sin}^{2} } + \frac{1 + {cos}^{2} }{1 + {cos}^{2} } \\ = 1 + 1 = 2$

RHS= 1+1 =2

hence proved