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Number of solution of the equation `cos^4 2x+2sin^2 2x=17(cosx+sinx)^8,0

Answer» `cos^4 2x+2sin^2 2x=17(cosx+sinx)^8`
`(cos^2 2x)^2+2(sin^2 2x)=17((cosx+sinx)^2)^4`
`(1-sin^2 2x)^2+2sin^2 2x=17(cos^2x+sin^2x+2sinxcosx)^4`
`1+sin^4 2x-2sin^2 2x+2sin^2 2x=17(1+sin2x)^4`
`1+sin^4 2x=17(1+sin2x)^4`
`1+y^2=1(1+y)^4`
Let` sin2x=y`
`1+y^4=17(1+y)^2(1+y)^2`
`(1+y^4)=17(1+2y+y^2)`
`1+y^4+17(1+4y^2+y^4+2y^2+4y^2+4y)`
`16y^4+102y^2+68y^3+68y+16=0`
`4y^2+17y^3+28y^2+17y+8=0`
`y=-1/2`
`sin2x=-1/2`
`2x=3pi+pi/6`
`x=7/12pi,11/12pi,19/12pi,23/12pi`.


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