If sin^-1x+sin^-1y+sin^-1z=pi,prove that 
(i) x(1-x^2)^1/2 +y(1-y^2)^1/2 +z(1-z^2)^1/2=2xyz
(ii) x^4 +y^4 +z^4 +4x^2 y^2 z^2 =2(x^2 y^2 +y^2 z^2+z^2-- x^2

Asked by Balbir | 28th Apr, 2017, 09:19: PM

Expert Answer:

begin mathsize 16px style Let comma space sin to the power of negative 1 end exponent straight x equals straight A comma space sin to the power of negative 1 end exponent straight y equals straight B comma space sin to the power of negative 1 end exponent straight z equals straight C
straight A plus straight B plus straight C equals straight pi
Here space to space prove space that space
sin space 2 straight A plus space sin 2 straight B plus sin 2 straight C equals 4 sinAsinBsinC
LHS
equals 2 sin left parenthesis straight A plus straight B right parenthesis cos left parenthesis straight A minus straight B right parenthesis plus 2 sinCcosC
equals 2 sinCcos left parenthesis straight A minus straight B right parenthesis plus 2 sinCcosC
equals 2 sinC left square bracket cos left parenthesis straight A minus straight B right parenthesis plus cosC right parenthesis right square bracket
equals 2 sinC 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
equals 2 sinC 2 sinBsinA
equals 4 sinAsinBsinC
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 sinAsinBsinC equals 2 xyz


end style
 
 
begin mathsize 16px style sin to the power of negative 1 end exponent straight x plus sin to the power of negative 1 end exponent straight y plus sin to the power of negative 1 end exponent straight z equals straight pi
sin to the power of negative 1 end exponent straight x plus sin to the power of negative 1 end exponent straight y equals straight pi minus sin to the power of negative 1 end exponent straight z
sin to the power of negative 1 end exponent open parentheses straight x square root of 1 minus straight y squared end root plus straight y square root of 1 minus straight x squared end root close parentheses equals open parentheses straight pi minus sin to the power of negative 1 end exponent straight z close parentheses
open parentheses straight x square root of 1 minus straight y squared end root plus straight y square root of 1 minus straight x squared end root close parentheses equals sin open parentheses straight pi minus sin to the power of negative 1 end exponent straight z close parentheses
straight x square root of 1 minus straight y squared end root plus straight y square root of 1 minus straight x squared end root equals straight z
Taking space square space on space both space sides space and space simplifying comma
straight x squared minus straight y squared minus straight z squared equals 2 yz square root of 1 minus straight x squared end root
squaring space on space both space sides comma
straight x to the power of 4 plus straight y to the power of 4 plus straight z to the power of 4 minus 2 straight x squared straight y squared plus 2 straight y squared straight z squared minus 2 straight z squared straight x squared equals 4 straight y squared straight z squared minus 4 straight x squared straight y squared straight z squared
straight x to the power of 4 space plus straight y to the power of 4 space plus straight z to the power of 4 space plus 4 straight x squared space straight y squared space straight z squared space equals 2 left parenthesis straight x squared space straight y squared space plus straight y squared space straight z squared plus straight z squared space straight x squared right parenthesis

end style

Answered by Sneha shidid | 2nd May, 2017, 10:14: AM