Question

Prove that:
\frac{cot A -cosA}{cot A + cos A} =\frac{cosecA-1}{cosecA+1}

Answer 1

[tex]\frac{cot A -cosA}{cot A + cos A} =\frac{cosecA-1}{cosecA+1} \\ L.H.S.=\ \textgreater \ \\ \frac{cot A -cosA}{cot A + cos A} \\ = \frac{ \frac{cosA}{sinA} -cosA}{ \frac{cosA} {sinA} + cos A} \\ = \frac{ \frac{cosA-cosA.sinA}{sinA} }{ \frac{cosA+cosA.sinA}{sinA}} \\ = \frac{ cosA(1-sinA) }{cosA(1+sinA)} = \frac{1-sinA}{1+sinA} \\ Dividing \: both \: numerator \ and \ denominator \ by \ sinA, \\ = \frac{ \frac{1-sinA}{sinA}}{ \frac{1+sinA}{sinA} } = \frac{ \frac{1}{sinA} -1 }{ \frac{1}{sinA} +1} [/tex]
$= \frac{cosecA-1}{cosecA+1} = R.H.S.$

Hence, proved.

Answer 2

It seems difficult but Om Gupta solved it in an easy way.