let −1≤x≤1 simplify A=sin{ arcsinx +2arcsin)(1−x)}


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Question Number 35416 by abdo mathsup 649 cc last updated on 18/May/18

$l e t - 1 ⩽ x ⩽ 1 s i m p l i f y$ $A = s i n {a r c s i n x + 2 a r c s i n) (1 - x)}$
Commented by abdo mathsup 649 cc last updated on 20/May/18

$w e h a v e$ $A = s i n (a r c s i n x) c o s (2 a r c s i n (1 - x))$ $+ c o s (a r c s i n x) s i n (2 a r c s i n (1 - x))$ $= x (1 - 2 {s i n}^{2} (a r c s i n (1 - x))) + c o s (a r c s i n x) s i n (a r c s i n (1 - x) c o s (a r c s i n (1 - x))$ $= x (1 - 2 {(1 - x)}^{2}) + (1 - x) c o s (a r c s i n x) c o s (a r c s i n (1 - x)) . . .$
Commented by abdo mathsup 649 cc last updated on 20/May/18
$but cos(arcsinx)=ξ(√(1−sin^2 (arcsinx))) =ξ(√(1−x^2 )) and cos(arcsin(1−x))=ξ^′ (√(1−(1−x)^2 )) so A=x{ 1−2(1−x)^2 }+ ξξ^′ (1−x)(√(1−x^2 )) (√(1−(1−x)^2 )) with ξ^2 =1 and ξ^′^2 =1$
$b u t c o s (a r c s i n x) = ξ \sqrt{1 - {s i n}^{2} (a r c s i n x)}$ $= ξ \sqrt{1 - x^{2}} a n d c o s (a r c s i n (1 - x)) = ξ^{^{'}} \sqrt{1 - {(1 - x)}^{2}}$ $s o$ $A = x {1 - 2 {(1 - x)}^{2}} + ξ ξ^{^{'}} (1 - x) \sqrt{1 - x^{2}} \sqrt{1 - {(1 - x)}^{2}}$ $w i t h ξ^{2} = 1 a n d ξ^{^{' 2}} = 1$
	Answered by tanmay.chaudhury50@gmail.com last updated on 18/May/18
	$sin(sin^(−1) x)cos{2sin^(−1) (1−x)}+cos(sin^(−1) x) ×sin{2sin^− (1−x)} x×(1−2sin^2 θ)+(√(1−x^2 )) ×2sinθ.(√(1−sin^2 θ)) sinθ=1−x =x×{1−2(1−2x+x^2 )+(√(1−x^2 )) ×2(1−x)× (√(1−(1−2x+x^2 ))) =x×(1−2+4x−2x^2 )+(√(1−x^2 )) ×(2−2x)× (√(1−1+2x−x^2 )) =x×(4x−2x^2 −1)+(√(1−x^2 )) ×(2−2x)(√(2x−2x^2 )) pls check$
	$s i n ({s i n}^{- 1} x) c o s {2 {s i n}^{- 1} (1 - x)} + c o s ({s i n}^{- 1} x)$ $\times s i n {2 {s i n}^{-} (1 - x)}$ $x \times (1 - 2 {s i n}^{2} θ) + \sqrt{1 - x^{2}} \times 2 s i n θ . \sqrt{1 - {s i n}^{2} θ}$ $s i n θ = 1 - x$ $= x \times {1 - 2 (1 - 2 x + x^{2}) + \sqrt{1 - x^{2}} \times 2 (1 - x) \times$ $\sqrt{1 - (1 - 2 x + x^{2})}$ $= x \times (1 - 2 + 4 x - 2 x^{2}) + \sqrt{1 - x^{2}} \times (2 - 2 x) \times$ $\sqrt{1 - 1 + 2 x - x^{2}}$ $= x \times (4 x - 2 x^{2} - 1) + \sqrt{1 - x^{2}} \times (2 - 2 x) \sqrt{2 x - 2 x^{2}}$ $p l s c h e c k$
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