let-1-x-1-simplify-A-sin-arcsinx-2arcsin-1-x-

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)) \dots

Commented by abdo mathsup 649 cc last updated on 20/May/18

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

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}}

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