निम्नलिखित त्रिकोणमितीय सर्वसमिकाओं को सिद्ध कीजिए :
`(i) (1 + sin theta)(1 - sin theta) = 1/(sec^(2) theta)" "(ii) cos^(2) theta(1+ tan^(2) theta) = 1`
(iii)` cos^(2) theta + 1/(1+cot^(2) theta) = 1 " "(iv) (1 + tan^(2) theta)(1 + sin theta)(1-sin theta) =1`
(v) ` (1 + cot^(2) theta)(1-cos theta)(1 + cos theta) = 1`

Explanation

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