If `sin^(4)x-cos^(4)x=p`, then which one of the following is correct ?

Options

A. p = 1
B. p = 0
C. `|p|gt1`
D. `|p|le1`

Answer

D

Explanation

Consider `sin^(4)x-cos^(4)x=p`
`rArr(sin^(2)x)^(2)-(cos^(2)x)^(2)=p`
`rArr(sin^(2)x-cos^(2)x(sin^(2)x+cos^(2)x)=p`
`rArrsin^(2)x-cos^(2)x=p(becausesin^(2)x+cos^(2)x=1)`
`rArr-cos2x=p(becausecos^(2)x-sin^(2)x=cos2x)`
`rArrcos2x=-p`
`therefore|p|le1`