Please solve for x. (sin x)^8+(cos x)^8=17/32

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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(`sinx)^8 + (cosx)^8 = 17/32 `

`(sin^4 x)^2 + (cos^4 x)^2 = 17/32 `

`==gt (sin^4 x + cos^4 x)^2 - 2sin^4 xcos^4 x = 17/32 `

`==gt ((sin^2 x+cos^2 x)^2 - 2sin^2 xcos^2 x)^2 - 2sin^4x cos^4 x = 17/32 `

`==gt (1-2sin^2x cos^2 x)^2 - 2sin^4 x cos^4 x = 17/32 `

`==gt 1-4sin^2 x cos^2 x + 4sin^4xcos^4 x - 2sin^4xcos^4 x = 17/32 `

`gt 1-4sin^2 x cos^2 x + 2sin^4xcos^4 x = 17/32`

`2(sinxcosx)^4 -4(sinxcosx)^2 + 1 = 17/32`

`==gt (2sinxcosx -1)^2 = 17/32 `

`==gt (sin2x -1)^2 = 17/32 `

`==gt sin2x -1 = +-0.7289 `

`==gt sin2x = 1+-0.7289`

`==gt sin2x = 1.7289 ==gt x= phi `

`==gt sin2x = 0.2711 ==gt 2x = 15.73 ==gt x = 7.87`

==> x = 7.87

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