Question

Prove 1- cos A /1+ cos A = (csc A – cotA)²

Answer 1

Prove:

`\sf ( 1- \cos A)/(1+ \cos A )= (\cosec A - \cot A)^2`

Let LHS =
`\sf ( 1- \cos A)/(1+ \cos A )`

RHS =
`\sf (\cosec A - \cot A)^2`

We need to Prove that LHS = RHS

Solving LHS:

`\sf ( 1- \cos A)/(1+ \cos A ) \\ \\ \sf (( 1- \cos A)(1- \cos A))/((1+ \cos A)(1- \cos A) ) \\ \\ \sf\frac{ { (1- \cos A)}^(2) }{ { 1- \cos }^(2)A } \\ \\ \sf \frac{{(1- \cos A)}^(2)} { \ { \sin}^(2) A} \\ \\ \sf \bigg( { (1 - \cos A)/(\sin A) } \bigg)^(2) \\ \\ \sf {\bigg( (1)/(\sin A) - (\cos A)/(\sin A) \bigg)}^(2) \\ \\ \sf {(\cosec A - \cot A)}^(2)`

LHS = RHS

Hence proved!

`{\boxed{\sf ( 1- \cos A)/(1+ \cos A )= (\cosec A - \cot A)^2}}`