Question

Prove that (sin theta +cos theta) (tan theta + cot theta) = sec theta +cosec theta

Answer 1

Answer:

See below

Step-by-step explanation:

$( \sin \theta + \cos \theta)( \tan \theta + \cot \theta) \\ = \sec \theta + \cosec \theta \\ \\ LHS = ( \sin \theta + \cos \theta)( \tan \theta + \cot \theta) \\ \\ = ( \sin \theta + \cos \theta) \bigg( \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta} \bigg) \\ \\ = ( \sin \theta + \cos \theta) \bigg( \frac{\sin ^{2} \theta +\cos ^{2} \theta }{\sin \theta \: \cos \theta} \bigg) \\ \\ = ( \sin \theta + \cos \theta) \bigg( \frac{ 1 }{\sin \theta \: \cos \theta} \bigg) \\ \\ = \frac{ \sin \theta + \cos \theta }{\sin \theta \: \cos \theta} \\ \\ = \frac{ \sin \theta }{\sin \theta \: \cos \theta} + \frac{\cos \theta }{\sin \theta \: \cos \theta}\\ \\ = \frac{ 1 }{\cos \theta} + \frac{1 }{\sin \theta} \\ \\ = \sec \theta + \cosec \theta \\ \\ = RHS$