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Question

Prove the following identities:

(1+cot A+tan A)(sin Acos A)=sec Acosec2Acosec Asec2A=sin A tan Acot A cos 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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