Prove each of the following identities :
` (i) sin^(6) theta + cos^(6)theta = 1- 3 sin^(2) theta cos^(2) theta `
`(ii) sin^(2)theta + cos^(4) theta = cos^(2) theta + sin^(4) theta `
`(iii) "cosec"^(4) theta - "cosec"^(2) theta = cot^(4) theta + cot^(2) theta `

Explanation

(i) `sin^(6)theta+cos^(6)theta=(sin^(2)theta)^(3)+(cos^(2)theta)^(3)`
`=(sin^(2)theta+cos^(2)theta)(sin^(4)theta+cos^(4)theta-sin^(2)thetacos^(2)theta)`
`=(sin^(2)theta+cos^(2)theta)^(2)-3 sin^(2)thetacos^(2)theta=(1-3sin^(2)thetacos^(2)theta).`
(ii) `(cos^(4)theta-sin^(4)theta)=(cos^(2)theta-sin^(2)theta)(cos^(2)theta+sin^(2)theta)`
`rArr (cos^(4)theta-sin^(4)theta)=(cos^(2)theta-sin^(2)theta)`
`rArr (sin^(2)theta+cos^(4)theta)=(cos^(2)theta+sin^(4)theta).`
(iii) `("cosec"^(4)theta-cot^(4)theta)=("cosec"^(2)theta+cot^(2)theta)("cosec"^(2)theta-cot^(2)theta) `
`rArr ("cosec"^(4)theta-cot^(4)theta)=("cosec"^(2)theta +cot^(2)theta) " " [ because "cosec"^(2)theta-cot^(2)theta=1]`
`rArr ("cosec"^(4)theta-"cosec"^(2)theta)=(cot^(4)theta+cot^(2)theta).`