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The number of distinct solutions of the ...

The number of distinct solutions of the equation `5/4cos^(2)2x + cos^4 x + sin^4 x+cos^6x+sin^6 x =2` in the interval `[0,2pi] ` is

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The correct Answer is:
8
`5/4 cos^(2) 2x+cos^(4)x+sin^(4) x+ cos^(6) x+sin^(6) x=2`
`rArr 5/4 cos^(2) 2x+(sin^(2) x+cos^(2) x)^(2)-2 sin^(2) x cos^(2) x + (sin^(2) x+cos^(2) x)^(3)-3 sin^(2) x cos^(2) x(sin^(2) x + cos^(2) x)=2`
`rArr 5/4 cos^(2) 2x+1-1/2 sin^(2) 2x+1-3/4 sin^(2) 2x=2`
`rArr cos^(2) 2x= sin^(2) 2x`
`rArr tan^(2) 2x=1 rArr tan 2x= pm 1`
`rArr 2x=npi pm pi/4, n in Z`
`rArr x=(4n pm 1) pi/8, n in Z`
`rArr x=pi/8, (3pi)/8, (5pi)/8, (7pi)/8, (9 pi)/8, (11pi)/8, (13pi)/8, (15 pi)/8`
So, number of solution `=8`.
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