How to simplify $arcsinleft( x sqrt 1-x - sqrtxsqrt1-x ^2right)$ without directly using an identity. [on hold]

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I know a solution that uses the identity
$$arcsin(x) + arcsin(y) =
\arcsinleft(x sqrt1−y^2 + y sqrt1−x^2right)$$



to simplify the expression
$$
arcsinleft( x sqrt 1-x - sqrtxsqrt1-x
^2right),\0<x<1
$$



but I wanted to know if there is any other method for simplifying this.




trigonometry trigonometric-series




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put on hold as off-topic by Andrés E. Caicedo, TheSimpliFire, Jendrik Stelzner, amWhy, Theoretical Economist yesterday


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, TheSimpliFire, Jendrik Stelzner, amWhy, Theoretical Economist
If this question can be reworded to fit the rules in the help center, please edit the question.












  • Hi William. Please edit your post to have a more descriptive title. Titles such as "Please do X for me" do not reflect the nature of the mathematical problem. Math.SE is meant to be a resource for mathematics, not a way to offshore work onto others, and your current title risks the latter interpretation. Thanks!
    – zahbaz
    Aug 28 at 16:51











  • @zahbaz, Then could you suggest me some appropriate title for this ??
    – William Wills
    Aug 29 at 1:05














up vote
-6
down vote

favorite












I know a solution that uses the identity
$$arcsin(x) + arcsin(y) =
\arcsinleft(x sqrt1−y^2 + y sqrt1−x^2right)$$



to simplify the expression
$$
arcsinleft( x sqrt 1-x - sqrtxsqrt1-x
^2right),\0<x<1
$$



but I wanted to know if there is any other method for simplifying this.




trigonometry trigonometric-series




share|cite|improve this question














put on hold as off-topic by Andrés E. Caicedo, TheSimpliFire, Jendrik Stelzner, amWhy, Theoretical Economist yesterday


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, TheSimpliFire, Jendrik Stelzner, amWhy, Theoretical Economist
If this question can be reworded to fit the rules in the help center, please edit the question.












  • Hi William. Please edit your post to have a more descriptive title. Titles such as "Please do X for me" do not reflect the nature of the mathematical problem. Math.SE is meant to be a resource for mathematics, not a way to offshore work onto others, and your current title risks the latter interpretation. Thanks!
    – zahbaz
    Aug 28 at 16:51











  • @zahbaz, Then could you suggest me some appropriate title for this ??
    – William Wills
    Aug 29 at 1:05












up vote
-6
down vote

favorite









up vote
-6
down vote

favorite











I know a solution that uses the identity
$$arcsin(x) + arcsin(y) =
\arcsinleft(x sqrt1−y^2 + y sqrt1−x^2right)$$



to simplify the expression
$$
arcsinleft( x sqrt 1-x - sqrtxsqrt1-x
^2right),\0<x<1
$$



but I wanted to know if there is any other method for simplifying this.




trigonometry trigonometric-series




share|cite|improve this question














I know a solution that uses the identity
$$arcsin(x) + arcsin(y) =
\arcsinleft(x sqrt1−y^2 + y sqrt1−x^2right)$$



to simplify the expression
$$
arcsinleft( x sqrt 1-x - sqrtxsqrt1-x
^2right),\0<x<1
$$



but I wanted to know if there is any other method for simplifying this.





trigonometry trigonometric-series





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 29 at 2:10









zahbaz

7,65521636




7,65521636










asked Aug 28 at 16:28









William Wills

1




1




put on hold as off-topic by Andrés E. Caicedo, TheSimpliFire, Jendrik Stelzner, amWhy, Theoretical Economist yesterday


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, TheSimpliFire, Jendrik Stelzner, amWhy, Theoretical Economist
If this question can be reworded to fit the rules in the help center, please edit the question.




put on hold as off-topic by Andrés E. Caicedo, TheSimpliFire, Jendrik Stelzner, amWhy, Theoretical Economist yesterday


This question appears to be off-topic. The users who voted to close gave this specific reason:


  • "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Andrés E. Caicedo, TheSimpliFire, Jendrik Stelzner, amWhy, Theoretical Economist
If this question can be reworded to fit the rules in the help center, please edit the question.











  • Hi William. Please edit your post to have a more descriptive title. Titles such as "Please do X for me" do not reflect the nature of the mathematical problem. Math.SE is meant to be a resource for mathematics, not a way to offshore work onto others, and your current title risks the latter interpretation. Thanks!
    – zahbaz
    Aug 28 at 16:51











  • @zahbaz, Then could you suggest me some appropriate title for this ??
    – William Wills
    Aug 29 at 1:05
















  • Hi William. Please edit your post to have a more descriptive title. Titles such as "Please do X for me" do not reflect the nature of the mathematical problem. Math.SE is meant to be a resource for mathematics, not a way to offshore work onto others, and your current title risks the latter interpretation. Thanks!
    – zahbaz
    Aug 28 at 16:51











  • @zahbaz, Then could you suggest me some appropriate title for this ??
    – William Wills
    Aug 29 at 1:05















Hi William. Please edit your post to have a more descriptive title. Titles such as "Please do X for me" do not reflect the nature of the mathematical problem. Math.SE is meant to be a resource for mathematics, not a way to offshore work onto others, and your current title risks the latter interpretation. Thanks!
– zahbaz
Aug 28 at 16:51





Hi William. Please edit your post to have a more descriptive title. Titles such as "Please do X for me" do not reflect the nature of the mathematical problem. Math.SE is meant to be a resource for mathematics, not a way to offshore work onto others, and your current title risks the latter interpretation. Thanks!
– zahbaz
Aug 28 at 16:51













@zahbaz, Then could you suggest me some appropriate title for this ??
– William Wills
Aug 29 at 1:05




@zahbaz, Then could you suggest me some appropriate title for this ??
– William Wills
Aug 29 at 1:05










1 Answer
1






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Assume that it's equal to $theta $ $$arcsin( x cdotsqrt 1-x - sqrtxcdotsqrt1-x
^2)=theta$$



$$ x cdotsqrt 1-x - sqrtxcdotsqrt1-x
^2=sin theta$$
$$ sin theta=sqrtxsqrt1-xleft(sqrtx -sqrt1+xright)$$
$$ sin theta=sqrtxsqrt1-xleft(sqrtx -sqrt1+xright)fracleft(sqrtx +sqrt1+xright)left(sqrtx +sqrt1+xright)$$
$$ sin theta=frac-sqrtxsqrt1-xleft(sqrtx +sqrt1+xright)$$






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  • but the answer is $arcsin(x)-arcsin(sqrtx)$
    – William Wills
    Aug 29 at 1:09


















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote













Assume that it's equal to $theta $ $$arcsin( x cdotsqrt 1-x - sqrtxcdotsqrt1-x
^2)=theta$$



$$ x cdotsqrt 1-x - sqrtxcdotsqrt1-x
^2=sin theta$$
$$ sin theta=sqrtxsqrt1-xleft(sqrtx -sqrt1+xright)$$
$$ sin theta=sqrtxsqrt1-xleft(sqrtx -sqrt1+xright)fracleft(sqrtx +sqrt1+xright)left(sqrtx +sqrt1+xright)$$
$$ sin theta=frac-sqrtxsqrt1-xleft(sqrtx +sqrt1+xright)$$






share|cite|improve this answer




















  • but the answer is $arcsin(x)-arcsin(sqrtx)$
    – William Wills
    Aug 29 at 1:09















up vote
0
down vote













Assume that it's equal to $theta $ $$arcsin( x cdotsqrt 1-x - sqrtxcdotsqrt1-x
^2)=theta$$



$$ x cdotsqrt 1-x - sqrtxcdotsqrt1-x
^2=sin theta$$
$$ sin theta=sqrtxsqrt1-xleft(sqrtx -sqrt1+xright)$$
$$ sin theta=sqrtxsqrt1-xleft(sqrtx -sqrt1+xright)fracleft(sqrtx +sqrt1+xright)left(sqrtx +sqrt1+xright)$$
$$ sin theta=frac-sqrtxsqrt1-xleft(sqrtx +sqrt1+xright)$$






share|cite|improve this answer




















  • but the answer is $arcsin(x)-arcsin(sqrtx)$
    – William Wills
    Aug 29 at 1:09













up vote
0
down vote










up vote
0
down vote









Assume that it's equal to $theta $ $$arcsin( x cdotsqrt 1-x - sqrtxcdotsqrt1-x
^2)=theta$$



$$ x cdotsqrt 1-x - sqrtxcdotsqrt1-x
^2=sin theta$$
$$ sin theta=sqrtxsqrt1-xleft(sqrtx -sqrt1+xright)$$
$$ sin theta=sqrtxsqrt1-xleft(sqrtx -sqrt1+xright)fracleft(sqrtx +sqrt1+xright)left(sqrtx +sqrt1+xright)$$
$$ sin theta=frac-sqrtxsqrt1-xleft(sqrtx +sqrt1+xright)$$






share|cite|improve this answer












Assume that it's equal to $theta $ $$arcsin( x cdotsqrt 1-x - sqrtxcdotsqrt1-x
^2)=theta$$



$$ x cdotsqrt 1-x - sqrtxcdotsqrt1-x
^2=sin theta$$
$$ sin theta=sqrtxsqrt1-xleft(sqrtx -sqrt1+xright)$$
$$ sin theta=sqrtxsqrt1-xleft(sqrtx -sqrt1+xright)fracleft(sqrtx +sqrt1+xright)left(sqrtx +sqrt1+xright)$$
$$ sin theta=frac-sqrtxsqrt1-xleft(sqrtx +sqrt1+xright)$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Aug 28 at 16:35









Deepesh Meena

3,1512824




3,1512824











  • but the answer is $arcsin(x)-arcsin(sqrtx)$
    – William Wills
    Aug 29 at 1:09

















  • but the answer is $arcsin(x)-arcsin(sqrtx)$
    – William Wills
    Aug 29 at 1:09
















but the answer is $arcsin(x)-arcsin(sqrtx)$
– William Wills
Aug 29 at 1:09





but the answer is $arcsin(x)-arcsin(sqrtx)$
– William Wills
Aug 29 at 1:09



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