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Prove that (i) (sin theta)/((1- cos th...

Prove that
`(i) (sin theta)/((1- cos theta))=("cosec" theta + cot theta) `
`(ii) (1)/((sec theta - tan theta))=(sec theta + tan theta) `

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We have
`(i) LHS= (sin theta)/((1- cos theta))`
`=(sin theta)/((1- cos theta)) xx ((1+ cos theta))/((1+ cos theta))`
[ multiplying num. and denom. by ` (1+ cos theta) ` ]
` =(sin theta(1+ cos theta))/((1- cos^(2)theta))=(sin theta(1+ cos theta))/(sin^(2)theta) = ((1+ cos theta))/(sin theta) `
`=((1)/(sin theta)+(cos theta)/(sin theta))=("cosec" theta + cot theta)= RHS. `
` therefore LHS = RHS. `
`(ii) LHS = (1)/((sec theta - tan theta)) `
` =(1)/((sec theta - tan theta)) xx ((sec theta + tan theta))/(( sec theta + tan theta))`
[ multiplying num. and denom. by `(sec theta + tan theta)`]
`=((sec theta + tan theta))/((sec^(2)theta - tan^(2)theta)) `
`=(sec theta + tan theta ) " " [ because sec^(2) theta- tan^(2) theta=1] `
`=RHS. `
` therefore LHS = RHS. `
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