Question

if x=sec(theta) +tan(theta),then x+1/x=

​

Answer 1

$x = \sec \alpha + \tan \alpha \\ x = \frac{1 + sin \alpha }{ \cos\alpha } \\ \frac{1}{x} = \frac{cos \alpha }{(1 + sin \alpha )} \\ x + \frac{1}{x} = \frac{(1 + sin \alpha }{ \cos \alpha } + \frac{cos \alpha }{(1 + \sin \alpha } \\ = \frac{(1 + {sin \alpha }^{2})2 + {cos}^{2} \alpha }{(1 + sin \alpha )cos \alpha } \\ \frac{1 + {sin}^{2} \alpha + 2sin \alpha + {cos}^{2} \alpha }{(1 + sin \alpha )cos \alpha } \: \: \: \: ( {sin}^{2} + {cos}^{2} = 1) \\ = \frac{2 + 2sin \alpha }{(1 + sin \alpha )cos \alpha } \\ = \frac{2(1 + sin \alpha )}{(1 + sin \alpha )cos \alpha } \\ \frac{2}{cos \alpha \\ } = 2sec \alpha \: ans$