Question

Prove that 1/(cosec A-cot A)-1/Sin A= 1/Sin A -1/(cosec A+cot A)​

Answer 1

$\displaystyle\huge\green{\underline{\underline{Solution}}}$

FORMULA TO BE IMPLEMENTED

${cosec }^{2}A - {cot }^{2} A= 1$

CALCULATION

LHS

$\displaystyle \: \frac{1}{ (cosec A-cot A)}- \frac{1}{Sin A}$

$= \displaystyle \: \frac{ (cosec A + cot A)}{ (cosec A + cot A)(cosec A-cot A)}- {cosec A}$

$= \displaystyle \: \frac{ ({cosec } A + cot A)}{ ({cosec }^{2}A - {cot }^{2} A)}- {cosec A}$

$= \displaystyle \: \frac{ ({cosec } A + cot A)}{ 1}- {cosec A}$

$= \displaystyle \: { ({cosec } A + cot A)}- {cosec A}$

$= cot A$

RHS

$\displaystyle \: \frac{1}{Sin A} - \frac{1}{ (cosec A + cot A)}$

$= \displaystyle \: {cosec A} - \frac{ (cosec A - cot A)}{ (cosec A + cot A)(cosec A-cot A)}$

$= \displaystyle \:{cosec A} - \frac{ ({cosec } A - cot A)}{ ({cosec }^{2}A - {cot }^{2} A)}$

$= \displaystyle \: {cosec A} - \frac{ ({cosec } A - cot A)}{ 1}$

$= \displaystyle \: {cosec A} - { ({cosec } A - cot A)}$

$= {cot A}$

Hence LHS = RHS

Hence proved

Answer 2

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Refer to the attachment