निम्नलिखित सर्वसमिकाओं को सिद्ध कीजिये :
(i) ` (cosec theta - cot theta)^(2) = (1-cos theta)/(1 + cos theta) " "(ii) cot theta - tan theta = (2 cos^(2) theta-1)/(sin theta * cos theta)`
(iii)` (tan theta+sec theta-1)/(tan theta - sec theta +1) = (1 + sin theta)/(cos theta)`

Explanation

(i) L.H.S. ` = (cosec theta - cot theta)^(2) `
` = (1/(sin theta ) - (cos theta )/(sin theta ))^(2) = ((1-cos theta )/(sin theta ))^(2) = ((1- cos theta )^(2))/(sin ^(2) theta ) = ((1-cos theta )^(2))/(1 - cos^(2) theta )`
` = ((1-cos theta)^(2))/((1- cos theta)(1+cos theta )) = (1- cos theta)/(1+ cos theta) = R.H.S.`
(ii) L.H.S. ` = cot theta - tan theta = (cos theta)/(sin theta ) - (sin theta)/(cos theta ) = (cos^(2) theta - sin^(2) theta )/(sin theta *cos theta)`
` = (cos^(2) theta - (1-cos^(2) theta))/(sin theta * cos theta) = (cos^(2) theta - 1 + cos ^(2) theta )/(sin theta * cos theta) `
` = (2 cos ^(2) theta -1)/(sin theta * cos theta) = R.H.S.`
(iii) L.H.S. ` (tan theta + sec theta - 1)/(tan theta - sec theta = 1) = (tan theta + sec theta - (sec^(2) theta - tan^(2) theta))/(tan theta - sec theta +1)`
` = (tan theta + sec theta - (sec theta - tan theta)(sec theta + tan theta))/(tan theta - sec theta + 1)`
` = ((tan theta + sec theta ) (1 - sec theta + tan theta ))/(tan theta - sec theta + 1) = tan theta + sec theta `
` = (sin theta)/(cos theta ) + 1/(cos theta ) = (sin theta +1)/(cos theta) = (1+ sin theta)/(cos theta) = R.H.S.`