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Given identity : (cos³x - cosx + sinx)/cosx = tanx - sin²x

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Start from left hand side

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= (cos³x - cosx + sinx)/cosx

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= cos³x/cosx- cosx/cosx + sinx/cosx

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= cos²x - 1 + tanx

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Substitute trigonometric identity : sin²x + cos²x = 1

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= cos²x - (sin²x + cos²x) + tanx

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= cos²x - sin²x - cos²x + tanx

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= - sin²x + tanx

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= tanx - sin²x

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Hence (cos³x - cosx + sinx)/cosx = tanx - sin²x proved

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