Question

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

Answer 1

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Answer 2

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First we take LHS

$= (\sin \alpha + cos \alpha)(tan \alpha + cot \alpha) \\ \\ = ( \sin \alpha + \cos \alpha )( \: \: \frac{ \sin \alpha }{ \: \: \: \cos \alpha } + \frac{ \cos \alpha }{ \sin \alpha } \: \: )$

= [sin (alpha) + cos (alpha)] [ (sin^2alpha+cos^alpha)/cos alpha × sin alpha ]

= [ sin alpha + cos alpha] [ 1/ (cos alpha ×sin alpha )]

= [ sin alpha + cos alpha] / [ sin alpha × cos alpha]


Now we take RHS

= sec alpha + cosec alpha

= 1/cos alpha + 1/sin alpha

= [ sin alpha + cos alpha] / [sin alpha × cos alpha ]


Now ,


LHS = RHS


HENCE PROVED