if sin^2 x+sin x = 1 what is cos^(12) x+3cos^(10) x+3cos^8 x+cos^6 x


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Question Number 78709 by jagoll last updated on 20/Jan/20

$if \sin^{2} x + \sin x = 1$ $what is \cos^{12} x + 3 \cos^{10} x + 3 \cos^{8} x + \cos^{6} x$
Commented by jagoll last updated on 20/Jan/20

$thanks sir john and sir som$
	Answered by john santu last updated on 20/Jan/20

	Commented by john santu last updated on 20/Jan/20
	$edit : cos^6 x=(((3−(√5))/2))×((((√5)−1)/2))=(√5)−2 ∴ ((√5)−2){(√5)−2+((9−3(√5)+3(√5)−3)/2)+1}= ((√5)−2)((√5)−1+(6/2)) = ((√5)−2)((√5)+2)=5−4 =1 ★$
	$e d i t : \cos^{6} x = (\frac{3 - \sqrt{5}}{2}) \times (\frac{\sqrt{5} - 1}{2}) = \sqrt{5} - 2$ $∴ (\sqrt{5} - 2) {\sqrt{5} - 2 + \frac{9 - 3 \sqrt{5} + 3 \sqrt{5} - 3}{2} + 1} =$ $(\sqrt{5} - 2) (\sqrt{5} - 1 + \frac{6}{2}) =$ $(\sqrt{5} - 2) (\sqrt{5} + 2) = 5 - 4 = 1 ★$
	Answered by som(math1967) last updated on 20/Jan/20

	${s i n}^{2} x + s i n x = 1$ $s i n x = 1 - {s i n}^{2} x = {c o s}^{2} x$ $∴ {s i n}^{2} x = {c o s}^{4} x$ $N o w {c o s}^{12} x + 3 {c o s}^{10} x + 3 {c o s}^{8} x + {c o s}^{6} x$ $= {({c o s}^{4} x)}^{3} + 3 {c o s}^{4} x . {c o s}^{2} x ({c o s}^{4} x + {c o s}^{2} x)$ $+ {({c o s}^{2} x)}^{3}$ $= {({c o s}^{4} x + {c o s}^{2} x)}^{3}$ $= {({s i n}^{2} x + {c o s}^{2} x)}^{3} = 1^{3} = 1 a n s ★$ $★ {c o s}^{4} x = {s i n}^{2} x f r o m c o n d i t i o n$
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