Question

If sin^–1'x + sin^–1 y + sin^–1 z = π. show that

$straight x square root of 1 minus straight x squared end root plus straight y square root of 1 minus straight y squared end root plus straight z square root of 1 minus straight z squared end root equals 2 space straight x space straight y space straight z$

Solution

We are given that
sin^–1x + sin–¹y+sin^–1z = π
Put x = sin A,y = sin B, z = sin C
∴ sin x = A, sin–1 y = B, sin^–1z = C
∴ we have
A + B + C = $straight pi$
$straight L. straight H. straight S. space equals space square root of 1 minus straight x squared end root plus straight y square root of 1 minus straight y to the power of 2 degree end exponent end root plus straight z square root of 1 minus straight z squared end root space space space space space space space space space space equals sin space straight A square root of 1 minus sin squared space straight A end root plus sin space straight B square root of 1 minus sin squared space straight B end root plus sin space straight C square root of 1 minus sin squared space straight C end root space space space space space space space space space space equals space sin space straight A. space cos space straight A space plus space sin space space straight B. space cos space straight B plus space sin space straight C space. space cos space straight C space space space space space space space space space space equals 1 half left square bracket 2 space sin space straight A. cos space straight A plus 2 space sin space straight B. space cos space straight B plus 2 space sin space straight C. space cos space straight C right square bracket space space space space space space space space space space equals 1 half left square bracket left parenthesis sin space 2 space straight A plus space sin space 2 space straight B right parenthesis plus sin space 2 space straight C right square bracket equals 1 half left square bracket 2 space sin space left parenthesis straight A plus straight B right parenthesis cos left parenthesis straight A minus straight B right parenthesis plus sin space 2 space straight C right square bracket space space space space space space space space space space equals space 1 half left square bracket 2 space sin left parenthesis 180 degree minus straight C right parenthesis space cos space left parenthesis straight A minus straight B right parenthesis plus sin space 2 straight C right square bracket space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket space space space space space space space space space space equals space 1 half space left square bracket 2 space sin space straight C. space cos space left parenthesis straight A minus straight B right parenthesis plus cos space straight C right curly bracket space space space space space space space space space space equals space sin space straight C left square bracket cos left parenthesis straight A minus straight B right parenthesis plus cos left curly bracket 180 degree minus left parenthesis straight A plus straight B right parenthesis right curly bracket right square bracket space space space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 1 right parenthesis right square bracket space space space space space space space space space space equals space sin space straight C left square bracket cos left parenthesis straight A minus straight B right parenthesis minus cos left parenthesis straight A plus straight B right parenthesis right square bracket minus sinC left square bracket 2 space sin space straight A space sin space straight B right square bracket space space space space space space space space space space equals space 2 space sin space straight A space sin space straight B space sin space straight C equals space 2 space xyz space space space space space space space space space space equals space straight R. straight H. straight S.$

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