प्रमाणित कीजिए कि `2 ( sin^(6)theta + cos^(6) theta)-3 ( sin^(4) theta + cos^(4) theta)+1 = 0 `

Explanation

L.H.S. `=2(sin^(6)theta+cos^(6)theta)-3(sin^(4)theta+cos^(4)theta)+1`
`=2[(sin^(2)theta)^(3)+(cos^(2)theta)^(3)]-3(sin^(4)theta+cos^(4)theta)+1`
`=2[(sin^(2)+cos^(2)theta)].[(sin^(2)theta)^(2)+(cos^(2)theta)^(2)-(sin^(2)thetacos^(2)theta)]`
`-3(sin^(4)theta+cos^(4)theta)+1" "[becausea^(3)+b^(3)=(a+b)(a^(2)+b^(2)-ab)]`
`=2[1.(sin^(4)theta+cos^(4)theta-sin^(2)thetacos^(2)theta)]-2(sin^(4)theta+cos^(4)theta+1)" "[becausesin^(2)theta+cos^(@)theta=1]`
`=2(sin^(4)theta+cos^(4)theta)-3sin^(2)thetacos^(2)theta-3(sin^(4)+cos^(4)theta)+1`
`=-(sin^(4)thetacos^(4)theta)-2sin^(2)thetacos^(2)theta+1`
`=-(sin^(4)theta+cos^(4)theta+2sin^(2)thetacos^(@)theta)+1`
`=-[(sin^(2)theta)^(2)+(cos^(2)theta)^(2)+2(sin^(2)theta)(cos^(2)theta)]+1`
`=-(sin^(2)theta+cos^(2)theta)^(2)+1" "[becausea^(2)+b^(2)+2ab=(a+b)^(2)]`
`=-(1)^(2)+1" "[becausesin^(2)theta+cos^(2)theta=1]`
`=-1+1=0`