We have

`(i)" LHS "= (1-sin^(2)theta)sec^(2) theta `

` = cos^(2)theta* sec^(2) theta " "[because 1-sin^(2)theta=cos^(2)theta ] `

` = cos^(2)theta * (1)/(cos^(2)theta)=1= " RHS." `

` therefore " LHS " = " RHS. " `

` (ii) " LHS " = (1+ tan^(2) theta )cos^(2) theta `

`= sec^(2)theta* cos^(2)theta " "[because 1+ tan^(2)theta=sec^(2)theta ]`

` = (1)/(cos^(2)theta)*cos^(2)theta=1="RHS." `

` therefore " LHS " = " RHS." `

` (iii) " LHS "= (1+ tan^(2)theta)(1-sin theta)(1+sin theta) `

` = (1+tan^(2) theta)(1-sin^(2)theta)= sec^(2) theta * cos^(2) theta `

` [ because (1+ tan^(2) theta )= sec^(2)theta , (1- sin^(2)theta )= cos^(2)theta] `

` =(1)/(cos^(2) theta)* cos^(2)theta " "[ because sec^(2) theta=(1)/(cos^(2)theta)] `

` =1= " RHS. " `

` therefore " LHS " = " RHS. " `

` (iv) " LHS "= 2 cos^(2) theta+(2)/((1+cot^(2) theta)) `

` =2 cos^(2) theta+(2)/("cosec"^(2)theta )" "[ because 1+cot^(2)theta="cosec"^(2) theta] `

`=2 cos^(2)theta+ 2sin^(2) theta " "[ because (1)/("cosec"^(2) theta)=sin^(2) theta] `

`=2(cos^(2)theta + sin^(2) theta)= 2 xx 1 " "[ because cos^(2) theta +sin^(2) theta=1 ] `

`= 2= "RHS." `

` therefore " LHS " = " RHS. " `