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Total number of solutions of sin^(4)x+c...

Total number of solutions of ` sin^(4)x+cos^(4)x=sinx*cosx` in `[0,2pi]` is equal to

A

2

B

4

C

6

D

8

Text Solution

Verified by Experts

The correct Answer is:
A
Given , `sin^(4)x+cos^(4)x=sinx*cosx`
`rArr(sin^(2)x+cos^(2)x)^(2)-2sin^(2)x*cos^(2)x=sinx*cosx`
`rArr1-(sin^(2)2x)/(2)=(sin2x)/(2)`
`rArrsin^(2)2x+sin2x-2=0`
`rArr(sin2x+2)(sin2x-1)=0`
`rArrsin2x=1 " "[becausesin2xge-1`, so sin 2x `ne-2]`
`therefore2x=(4n+1)(pi)/(2)`
`rArrx=(4n+1)(pi)/(4)`
`rArrx=(pi)/(4),(5pi)/(4)` (at n=0 and1)
Hence, two solutions exist.
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