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Question Number 196929 by Amidip last updated on 03/Sep/23

	Answered by som(math1967) last updated on 03/Sep/23
	$2U_6 −3U_4 +1 =2(sin^6 α+cos^6 α)−3(sin^4 α+cos^4 )+1 =2{(sin^2 α+cos^2 α)^3 −3sin^2 αcos^2 α(sin^2 α+cos^2 α) −3{(sin^2 α+cos^2 α)^2 −2sin^2 αcos^2 α}+1 =2−6sin^2 αcos^2 α−3+6sin^2 αcos^2 α+1 =0$
	$2 U_{6} - 3 U_{4} + 1$ $= 2 ({s i n}^{6} α + {c o s}^{6} α) - 3 ({s i n}^{4} α + {c o s}^{4}) + 1$ $= 2 {{({s i n}^{2} α + {c o s}^{2} α)}^{3} - 3 {s i n}^{2} α {c o s}^{2} α ({s i n}^{2} α + {c o s}^{2} α)$ $- 3 {{({s i n}^{2} α + {c o s}^{2} α)}^{2} - 2 {s i n}^{2} α {c o s}^{2} α} + 1$ $= 2 - 6 {s i n}^{2} α {c o s}^{2} α - 3 + 6 {s i n}^{2} α {c o s}^{2} α + 1$ $= 0$
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