Question

prove: (secthita + tanthita -1)(secthita- tanthita +1) = 2 tanthita​

Answer 1

$(sec \theta + tan \theta - 1)(sec \theta - tan \theta + 1) \\ = ( \frac{1 + sin \theta - cos \theta}{cos \theta} )( \frac{1 - sin \theta + co s \theta}{cos \theta} ) \\ = ( \frac{1 + sin \theta - cos \theta}{cos \theta} )( \frac{1 - (sin \theta - cos \theta)}{cos \theta} ) \\ = \frac{1 - ( {sin \theta - cos \theta})^{2} }{ {cos}^{2} \theta } \\ = \frac{1 - 1 + 2sin \theta \: cos \theta}{ {cos}^{2} \theta } \\ = \frac{2sin \theta \: cos \theta}{ {cos}^{2} \theta} \\ = 2 \frac{sin \theta}{co s \theta} \\ = 2tan \theta$