Question

if sin theta + cos theta=X then sin theta - cos theta =?​

Answer 1

$\: sin \theta + cos \theta = x \\ ( {sin \theta + cos \theta})^{2} = {x}^{2} \\ 1 + 2sin \theta cos \theta = {x}^{2} \\ sin \theta cos \theta = \frac{ {x}^{2} - 1}{2} \\ \\ \: now \: \: \: \: sin \theta - cos \theta = \sqrt{( {sin \theta + cos \theta)}^{2} - 4sin \theta cos \theta} \\ = \sqrt{ {x}^{2} - 4( \frac{ {x }^{2} - 1}{2}) } \\ = \sqrt{ {x}^{2} - 2 {x}^{2} + 2} \\ = \: \sqrt{2 - {x}^{2} }$