Question

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

Answer 1

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Answer 2

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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( (\sin \theta)/(\cos \theta) + (\cos \theta)/(\sin \theta) \bigg) \\ \\ = ( \sin \theta + \cos \theta) \bigg( (\sin ^(2) \theta +\cos ^(2) \theta )/(\sin \theta \: \cos \theta) \bigg) \\ \\ = ( \sin \theta + \cos \theta) \bigg( ( 1 )/(\sin \theta \: \cos \theta) \bigg) \\ \\ = ( \sin \theta + \cos \theta )/(\sin \theta \: \cos \theta) \\ \\ = ( \sin \theta )/(\sin \theta \: \cos \theta) + (\cos \theta )/(\sin \theta \: \cos \theta)\\ \\ = ( 1 )/(\cos \theta) + (1 )/(\sin \theta) \\ \\ = \sec \theta + \cosec \theta \\ \\ = RHS`