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Trigonometric Identities

Prove the fol...

Question

Prove the following identities:

$(1 + c o t A + t a n A) (s i n A - c o s A) = \frac{s e c A}{c o s e c^{2} A} - \frac{c o s e c A}{s e c^{2} A} = s i n A t a n A - c o t A c o s A$

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Solution

L.H.S
$equals left parenthesis 1 plus fraction numerator cos A over denominator sin space A end fraction plus fraction numerator sin space A over denominator cos space A end fraction right parenthesis left parenthesis s i n A minus c o s A right parenthesis space equals left parenthesis fraction numerator cos A sin A plus cos squared A plus S i n squared A over denominator cos A space sin A end fraction right parenthesis left parenthesis s i n A minus cos A right parenthesis equals left parenthesis fraction numerator cos A sin A plus 1 over denominator cos A space sin A end fraction right parenthesis left parenthesis s i n A minus cos A right parenthesis equals fraction numerator cos A sin squared A minus cos squared A sin A plus sin A minus cos A over denominator cos space A space sin space A end fraction equals fraction numerator sin A left parenthesis 1 minus cos squared A right parenthesis minus cos A left parenthesis 1 minus sin squared A right parenthesis over denominator cos space A space sin space A end fraction equals fraction numerator sin cubed A minus cos cubed A over denominator cos space A space sin space A end fraction equals fraction numerator sin cubed A over denominator sin space A space cos space A end fraction minus fraction numerator cos cubed A over denominator sin space A space cos space A end fraction equals fraction numerator sin squared A over denominator cos space A end fraction minus fraction numerator cos squared over denominator sin space A end fraction equals fraction numerator s e c A over denominator cos e c space squared A end fraction minus fraction numerator cos e c space A over denominator s e c squared A end fraction equals sin space A space tan space A space minus space c o t space A space cos space A$
HENCE PROVED

L.H.S = R.H.S

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(1 + \cot A + \tan A) (\sin A - \cos A) = \frac{\sec A}{{cosec}^{2} A} - \frac{cosec A}{\sec^{2} A} = \sin A \tan A - \cot A \cos A

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