MHB Solving Trig Identity: Sin[1/2 sin^−1 (x)] = 1/2 X (sqr(1+x) –sqr(1-x))

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The discussion focuses on solving the trigonometric identity sin[1/2 sin^−1(x)] = 1/2 * (sqrt(1+x) - sqrt(1-x)). The original poster attempts to manipulate the equation by letting 2y = sin^−1(x) and using the identity x = sin(2y). They express x in terms of sine and cosine but struggle to progress further. The request for clarity emphasizes avoiding the use of 'x' or 'X' in favor of alternative notations. The conversation highlights the challenges in simplifying and proving the given trigonometric identity.
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Good morning, may I possibly get a clue into solving this problem : check this identity: sin[1/2 sin^−1 (x) ]=1/2 X (sqr(1+x) –sqr(1-x)). I tried to say "2y=sin^−1 (x)", Then f=sin(y), then "x=sin(2y)", x=2sin(x) X cos(x), but I can't find further. Thank you very much.
 
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DUC_123 said:
Good morning, may I possibly get a clue into solving this problem : check this identity: sin[1/2 sin^−1 (x) ]=1/2 X (sqr(1+x) –sqr(1-x)). I tried to say "2y=sin^−1 (x)", Then f=sin(y), then "x=sin(2y)", x=2sin(y) X cos(y), but I can't find further. Thank you very much.
 
Please, no x's or X's. If you must use something use *. Many people just use a space, such as in x = 2 sin(y) cos(y).

-Dan
 

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