निम्नलिखित सर्वसमिकाओं को सिद्ध कीजिए
`(i) 1 + (cot^(2) theta)/(1+cosec theta) = cosec theta " "(ii) sec^(4) theta - sec^(2) theta = tan^(4) theta + tan^(2) theta`
(iii)` 2 sec^(2) theta - sec^(4) theta - 2 cosec^(2) theta + cosec^(4) theta ` `= cot^(4) theta - tan^(4) theta`

Explanation

(i) L.H.S. ` = 1 + (cot^(2) theta)/(1+cosec theta) = 1 + (cosec^(2) theta -1)/(1 + cosec theta)`
` = 1 + ((cosec theta + 1)(cosec theta - 1))/(1 + cosec theta) = 1 + cosec theta - 1`
` = cosec theta = R.H.S.`
(ii) L.H.S. ` = sec^(4) theta - sec^(2) theta = sec^(2) theta ( sec^(2) theta - 1)`
` = (1 + tan^(2) theta ) tan^(2) theta = tan^(2) theta + tan^(4) theta`
` = tan^(4) theta + tan^(2) theta = R.H.S.`
(iii) L.H.S. ` = 2 sec^(2) theta - sec^(4) theta - 2 cosec^(2) theta + cosec^(4) theta `
` = 2 sec^(2) theta - 2 cosec^(2) theta - sec^(4) theta + cosec^(4) theta`
` = 2 (sec^(2) theta - cosec^(2) theta) - (sec^(4) theta - cosec^(4)theta)`
` = 2 (sec^(2) theta - cosec^(2) theta)-(sec^(2) theta - cosec^(2) theta) xx (sec^(2) theta + cosec^(2) theta)`
` = (sec^(2) theta - cosec^(2) theta)(2 - sec^(2) theta - cosec^(2) theta)`
` = (1 + tan^(2) theta - 1 - cot^(2) theta)(2-1- tan^(2) theta -1 - cot^(2) theta)`
` = (tan^(2) theta - cot^(2) theta)(-tan^(2) theta - cot^(2) theta)`
` = - (tan^(2) theta - cot^(2) theta)(tan^(2) theta +cot^(2) theta)`
` = - (tan^(4) theta - cot^(4) theta)` ltBrgt ` = cot^(4) theta - tan^(4) theta = R.H.S.`