Prove $cot^{-1}left(frac{sqrt{1+sin x}+sqrt{1-sin x}}{sqrt{1+sin x}-sqrt{1-sin x}}right)=frac x2$











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I have the following question:




Prove that: $$ cot^{-1}Biggl(frac{sqrt{1+sin x} + sqrt{1-sin x}}{sqrt{1+sin x} - sqrt{1-sin x}}Biggl) = frac x2, x in biggl(0, frac pi4biggl) $$



The solution:



enter image description here




My question is about the second step (i.e., highlighted). We know that it has came from the following resolution: $$ sqrt{1pm sin x} = sqrt{sin^2frac x 2 + cos^2frac x 2 pm2sinfrac x 2cosfrac x 2} = pm biggl( cosfrac x 2 pm sinfrac x 2 biggl) $$



I've included the $pm$ symbol in accordance with the standard definition of square root. The above solution uses the positive resolution (i.e., $ cosfrac x 2 - sinfrac x 2 $) for $ sqrt{1- sin x} $. But, if I use the negative one, then the result changes. The sixth step changes to: $$ cot^{-1}biggl(tanfrac x 2biggl) = cot^{-1}Biggl(cotleft(frac pi 2 - frac x 2right)Biggl) $$



which yield the result: $$ frac pi 2 - frac x 2 $$



Mathematically, this result is different from that provided in the RHS of question.



Is the question statement wrong or I've been hacked up?










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  • @Winther Ok, I edited the question.
    – rv7
    Dec 1 at 13:42










  • The range $xin(0,pi/4)$ has to be kept in mind while removing the square root.
    – StubbornAtom
    Dec 1 at 13:55










  • It’s already given x belong to 0 to pi/4. That means x is positive. Which also means sinx + cos is positive
    – Fawad
    2 days ago















up vote
2
down vote

favorite












I have the following question:




Prove that: $$ cot^{-1}Biggl(frac{sqrt{1+sin x} + sqrt{1-sin x}}{sqrt{1+sin x} - sqrt{1-sin x}}Biggl) = frac x2, x in biggl(0, frac pi4biggl) $$



The solution:



enter image description here




My question is about the second step (i.e., highlighted). We know that it has came from the following resolution: $$ sqrt{1pm sin x} = sqrt{sin^2frac x 2 + cos^2frac x 2 pm2sinfrac x 2cosfrac x 2} = pm biggl( cosfrac x 2 pm sinfrac x 2 biggl) $$



I've included the $pm$ symbol in accordance with the standard definition of square root. The above solution uses the positive resolution (i.e., $ cosfrac x 2 - sinfrac x 2 $) for $ sqrt{1- sin x} $. But, if I use the negative one, then the result changes. The sixth step changes to: $$ cot^{-1}biggl(tanfrac x 2biggl) = cot^{-1}Biggl(cotleft(frac pi 2 - frac x 2right)Biggl) $$



which yield the result: $$ frac pi 2 - frac x 2 $$



Mathematically, this result is different from that provided in the RHS of question.



Is the question statement wrong or I've been hacked up?










share|cite|improve this question
























  • @Winther Ok, I edited the question.
    – rv7
    Dec 1 at 13:42










  • The range $xin(0,pi/4)$ has to be kept in mind while removing the square root.
    – StubbornAtom
    Dec 1 at 13:55










  • It’s already given x belong to 0 to pi/4. That means x is positive. Which also means sinx + cos is positive
    – Fawad
    2 days ago













up vote
2
down vote

favorite









up vote
2
down vote

favorite











I have the following question:




Prove that: $$ cot^{-1}Biggl(frac{sqrt{1+sin x} + sqrt{1-sin x}}{sqrt{1+sin x} - sqrt{1-sin x}}Biggl) = frac x2, x in biggl(0, frac pi4biggl) $$



The solution:



enter image description here




My question is about the second step (i.e., highlighted). We know that it has came from the following resolution: $$ sqrt{1pm sin x} = sqrt{sin^2frac x 2 + cos^2frac x 2 pm2sinfrac x 2cosfrac x 2} = pm biggl( cosfrac x 2 pm sinfrac x 2 biggl) $$



I've included the $pm$ symbol in accordance with the standard definition of square root. The above solution uses the positive resolution (i.e., $ cosfrac x 2 - sinfrac x 2 $) for $ sqrt{1- sin x} $. But, if I use the negative one, then the result changes. The sixth step changes to: $$ cot^{-1}biggl(tanfrac x 2biggl) = cot^{-1}Biggl(cotleft(frac pi 2 - frac x 2right)Biggl) $$



which yield the result: $$ frac pi 2 - frac x 2 $$



Mathematically, this result is different from that provided in the RHS of question.



Is the question statement wrong or I've been hacked up?










share|cite|improve this question















I have the following question:




Prove that: $$ cot^{-1}Biggl(frac{sqrt{1+sin x} + sqrt{1-sin x}}{sqrt{1+sin x} - sqrt{1-sin x}}Biggl) = frac x2, x in biggl(0, frac pi4biggl) $$



The solution:



enter image description here




My question is about the second step (i.e., highlighted). We know that it has came from the following resolution: $$ sqrt{1pm sin x} = sqrt{sin^2frac x 2 + cos^2frac x 2 pm2sinfrac x 2cosfrac x 2} = pm biggl( cosfrac x 2 pm sinfrac x 2 biggl) $$



I've included the $pm$ symbol in accordance with the standard definition of square root. The above solution uses the positive resolution (i.e., $ cosfrac x 2 - sinfrac x 2 $) for $ sqrt{1- sin x} $. But, if I use the negative one, then the result changes. The sixth step changes to: $$ cot^{-1}biggl(tanfrac x 2biggl) = cot^{-1}Biggl(cotleft(frac pi 2 - frac x 2right)Biggl) $$



which yield the result: $$ frac pi 2 - frac x 2 $$



Mathematically, this result is different from that provided in the RHS of question.



Is the question statement wrong or I've been hacked up?







trigonometry proof-verification






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edited 2 days ago

























asked Dec 1 at 12:45









rv7

1339




1339












  • @Winther Ok, I edited the question.
    – rv7
    Dec 1 at 13:42










  • The range $xin(0,pi/4)$ has to be kept in mind while removing the square root.
    – StubbornAtom
    Dec 1 at 13:55










  • It’s already given x belong to 0 to pi/4. That means x is positive. Which also means sinx + cos is positive
    – Fawad
    2 days ago


















  • @Winther Ok, I edited the question.
    – rv7
    Dec 1 at 13:42










  • The range $xin(0,pi/4)$ has to be kept in mind while removing the square root.
    – StubbornAtom
    Dec 1 at 13:55










  • It’s already given x belong to 0 to pi/4. That means x is positive. Which also means sinx + cos is positive
    – Fawad
    2 days ago
















@Winther Ok, I edited the question.
– rv7
Dec 1 at 13:42




@Winther Ok, I edited the question.
– rv7
Dec 1 at 13:42












The range $xin(0,pi/4)$ has to be kept in mind while removing the square root.
– StubbornAtom
Dec 1 at 13:55




The range $xin(0,pi/4)$ has to be kept in mind while removing the square root.
– StubbornAtom
Dec 1 at 13:55












It’s already given x belong to 0 to pi/4. That means x is positive. Which also means sinx + cos is positive
– Fawad
2 days ago




It’s already given x belong to 0 to pi/4. That means x is positive. Which also means sinx + cos is positive
– Fawad
2 days ago










2 Answers
2






active

oldest

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up vote
1
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Rationalize the numerator to find



$$f(x)=dfrac{1+|cos x|}{sin x}$$



Now if $cos xge0,$ $$f(x)=cotdfrac x2$$



$cot^{-1}f(x)=?$



Else $f(x)=dfrac{1-cos x}{sin x}=tandfrac x2$



Now $cot^{-1}f(x)=dfracpi2-tan^{-1}f(x)=?$






share|cite|improve this answer




























    up vote
    1
    down vote













    As StubbornAtom mentioned in comment, for $x in biggl(0, frac pi4biggl)$, the square root is:
    $$sqrt{1-sin x}=cos frac x2-sin frac x2,$$
    because:
    $$cos frac x2-sin frac x2>0.$$
    If, for example, $x in biggl(frac pi2, pibiggl)$, the square root would be:
    $$sqrt{1-sin x}=sinfrac x2-cos frac x2,$$
    because:
    $$sin frac x2-cos frac x2>0.$$



    Addendum. Note that $pm$ is placed to make the number positive and it does not imply one gets positive and negative values for the square root. Here is another way to write the same equality:
    $$sqrt{1-sin x}=left|cos frac x2-sin frac x2right|=begin{cases}+left(cos frac x2-sin frac x2right), cos frac x2-sin frac x2ge 0 \ -left(cos frac x2-sin frac x2right), cos frac x2-sin frac x2< 0 \ end{cases}.$$






    share|cite|improve this answer























    • Why not $sqrt{1-sin x}= - bigl( cos frac x2-sin frac x2 bigl)$? Because $sqrt9 = pm3$
      – rv7
      Dec 2 at 2:28










    • Firstly, $0le 1-sin xle 2<9$. Secondly, $sqrt{9}=3$ and $pm sqrt{9}=pm 3$. Thirdly, $0<x<pi/4$. For example, when $x=pi/6$, $sqrt{1-sin (pi/6)}=1/sqrt{2}=cos (pi/12)-sin (pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$. Fourthly, for $x=5pi/6>pi/2>pi/4$, we get what you are saying: $sqrt{1-sin (5pi/6)}=1/sqrt{2}=sin (5pi/12)-cos (5pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$.
      – farruhota
      2 days ago













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    2 Answers
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    up vote
    1
    down vote













    Rationalize the numerator to find



    $$f(x)=dfrac{1+|cos x|}{sin x}$$



    Now if $cos xge0,$ $$f(x)=cotdfrac x2$$



    $cot^{-1}f(x)=?$



    Else $f(x)=dfrac{1-cos x}{sin x}=tandfrac x2$



    Now $cot^{-1}f(x)=dfracpi2-tan^{-1}f(x)=?$






    share|cite|improve this answer

























      up vote
      1
      down vote













      Rationalize the numerator to find



      $$f(x)=dfrac{1+|cos x|}{sin x}$$



      Now if $cos xge0,$ $$f(x)=cotdfrac x2$$



      $cot^{-1}f(x)=?$



      Else $f(x)=dfrac{1-cos x}{sin x}=tandfrac x2$



      Now $cot^{-1}f(x)=dfracpi2-tan^{-1}f(x)=?$






      share|cite|improve this answer























        up vote
        1
        down vote










        up vote
        1
        down vote









        Rationalize the numerator to find



        $$f(x)=dfrac{1+|cos x|}{sin x}$$



        Now if $cos xge0,$ $$f(x)=cotdfrac x2$$



        $cot^{-1}f(x)=?$



        Else $f(x)=dfrac{1-cos x}{sin x}=tandfrac x2$



        Now $cot^{-1}f(x)=dfracpi2-tan^{-1}f(x)=?$






        share|cite|improve this answer












        Rationalize the numerator to find



        $$f(x)=dfrac{1+|cos x|}{sin x}$$



        Now if $cos xge0,$ $$f(x)=cotdfrac x2$$



        $cot^{-1}f(x)=?$



        Else $f(x)=dfrac{1-cos x}{sin x}=tandfrac x2$



        Now $cot^{-1}f(x)=dfracpi2-tan^{-1}f(x)=?$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 1 at 13:11









        lab bhattacharjee

        221k15155272




        221k15155272






















            up vote
            1
            down vote













            As StubbornAtom mentioned in comment, for $x in biggl(0, frac pi4biggl)$, the square root is:
            $$sqrt{1-sin x}=cos frac x2-sin frac x2,$$
            because:
            $$cos frac x2-sin frac x2>0.$$
            If, for example, $x in biggl(frac pi2, pibiggl)$, the square root would be:
            $$sqrt{1-sin x}=sinfrac x2-cos frac x2,$$
            because:
            $$sin frac x2-cos frac x2>0.$$



            Addendum. Note that $pm$ is placed to make the number positive and it does not imply one gets positive and negative values for the square root. Here is another way to write the same equality:
            $$sqrt{1-sin x}=left|cos frac x2-sin frac x2right|=begin{cases}+left(cos frac x2-sin frac x2right), cos frac x2-sin frac x2ge 0 \ -left(cos frac x2-sin frac x2right), cos frac x2-sin frac x2< 0 \ end{cases}.$$






            share|cite|improve this answer























            • Why not $sqrt{1-sin x}= - bigl( cos frac x2-sin frac x2 bigl)$? Because $sqrt9 = pm3$
              – rv7
              Dec 2 at 2:28










            • Firstly, $0le 1-sin xle 2<9$. Secondly, $sqrt{9}=3$ and $pm sqrt{9}=pm 3$. Thirdly, $0<x<pi/4$. For example, when $x=pi/6$, $sqrt{1-sin (pi/6)}=1/sqrt{2}=cos (pi/12)-sin (pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$. Fourthly, for $x=5pi/6>pi/2>pi/4$, we get what you are saying: $sqrt{1-sin (5pi/6)}=1/sqrt{2}=sin (5pi/12)-cos (5pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$.
              – farruhota
              2 days ago

















            up vote
            1
            down vote













            As StubbornAtom mentioned in comment, for $x in biggl(0, frac pi4biggl)$, the square root is:
            $$sqrt{1-sin x}=cos frac x2-sin frac x2,$$
            because:
            $$cos frac x2-sin frac x2>0.$$
            If, for example, $x in biggl(frac pi2, pibiggl)$, the square root would be:
            $$sqrt{1-sin x}=sinfrac x2-cos frac x2,$$
            because:
            $$sin frac x2-cos frac x2>0.$$



            Addendum. Note that $pm$ is placed to make the number positive and it does not imply one gets positive and negative values for the square root. Here is another way to write the same equality:
            $$sqrt{1-sin x}=left|cos frac x2-sin frac x2right|=begin{cases}+left(cos frac x2-sin frac x2right), cos frac x2-sin frac x2ge 0 \ -left(cos frac x2-sin frac x2right), cos frac x2-sin frac x2< 0 \ end{cases}.$$






            share|cite|improve this answer























            • Why not $sqrt{1-sin x}= - bigl( cos frac x2-sin frac x2 bigl)$? Because $sqrt9 = pm3$
              – rv7
              Dec 2 at 2:28










            • Firstly, $0le 1-sin xle 2<9$. Secondly, $sqrt{9}=3$ and $pm sqrt{9}=pm 3$. Thirdly, $0<x<pi/4$. For example, when $x=pi/6$, $sqrt{1-sin (pi/6)}=1/sqrt{2}=cos (pi/12)-sin (pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$. Fourthly, for $x=5pi/6>pi/2>pi/4$, we get what you are saying: $sqrt{1-sin (5pi/6)}=1/sqrt{2}=sin (5pi/12)-cos (5pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$.
              – farruhota
              2 days ago















            up vote
            1
            down vote










            up vote
            1
            down vote









            As StubbornAtom mentioned in comment, for $x in biggl(0, frac pi4biggl)$, the square root is:
            $$sqrt{1-sin x}=cos frac x2-sin frac x2,$$
            because:
            $$cos frac x2-sin frac x2>0.$$
            If, for example, $x in biggl(frac pi2, pibiggl)$, the square root would be:
            $$sqrt{1-sin x}=sinfrac x2-cos frac x2,$$
            because:
            $$sin frac x2-cos frac x2>0.$$



            Addendum. Note that $pm$ is placed to make the number positive and it does not imply one gets positive and negative values for the square root. Here is another way to write the same equality:
            $$sqrt{1-sin x}=left|cos frac x2-sin frac x2right|=begin{cases}+left(cos frac x2-sin frac x2right), cos frac x2-sin frac x2ge 0 \ -left(cos frac x2-sin frac x2right), cos frac x2-sin frac x2< 0 \ end{cases}.$$






            share|cite|improve this answer














            As StubbornAtom mentioned in comment, for $x in biggl(0, frac pi4biggl)$, the square root is:
            $$sqrt{1-sin x}=cos frac x2-sin frac x2,$$
            because:
            $$cos frac x2-sin frac x2>0.$$
            If, for example, $x in biggl(frac pi2, pibiggl)$, the square root would be:
            $$sqrt{1-sin x}=sinfrac x2-cos frac x2,$$
            because:
            $$sin frac x2-cos frac x2>0.$$



            Addendum. Note that $pm$ is placed to make the number positive and it does not imply one gets positive and negative values for the square root. Here is another way to write the same equality:
            $$sqrt{1-sin x}=left|cos frac x2-sin frac x2right|=begin{cases}+left(cos frac x2-sin frac x2right), cos frac x2-sin frac x2ge 0 \ -left(cos frac x2-sin frac x2right), cos frac x2-sin frac x2< 0 \ end{cases}.$$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited 2 days ago

























            answered Dec 1 at 16:16









            farruhota

            18.1k2736




            18.1k2736












            • Why not $sqrt{1-sin x}= - bigl( cos frac x2-sin frac x2 bigl)$? Because $sqrt9 = pm3$
              – rv7
              Dec 2 at 2:28










            • Firstly, $0le 1-sin xle 2<9$. Secondly, $sqrt{9}=3$ and $pm sqrt{9}=pm 3$. Thirdly, $0<x<pi/4$. For example, when $x=pi/6$, $sqrt{1-sin (pi/6)}=1/sqrt{2}=cos (pi/12)-sin (pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$. Fourthly, for $x=5pi/6>pi/2>pi/4$, we get what you are saying: $sqrt{1-sin (5pi/6)}=1/sqrt{2}=sin (5pi/12)-cos (5pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$.
              – farruhota
              2 days ago




















            • Why not $sqrt{1-sin x}= - bigl( cos frac x2-sin frac x2 bigl)$? Because $sqrt9 = pm3$
              – rv7
              Dec 2 at 2:28










            • Firstly, $0le 1-sin xle 2<9$. Secondly, $sqrt{9}=3$ and $pm sqrt{9}=pm 3$. Thirdly, $0<x<pi/4$. For example, when $x=pi/6$, $sqrt{1-sin (pi/6)}=1/sqrt{2}=cos (pi/12)-sin (pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$. Fourthly, for $x=5pi/6>pi/2>pi/4$, we get what you are saying: $sqrt{1-sin (5pi/6)}=1/sqrt{2}=sin (5pi/12)-cos (5pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$.
              – farruhota
              2 days ago


















            Why not $sqrt{1-sin x}= - bigl( cos frac x2-sin frac x2 bigl)$? Because $sqrt9 = pm3$
            – rv7
            Dec 2 at 2:28




            Why not $sqrt{1-sin x}= - bigl( cos frac x2-sin frac x2 bigl)$? Because $sqrt9 = pm3$
            – rv7
            Dec 2 at 2:28












            Firstly, $0le 1-sin xle 2<9$. Secondly, $sqrt{9}=3$ and $pm sqrt{9}=pm 3$. Thirdly, $0<x<pi/4$. For example, when $x=pi/6$, $sqrt{1-sin (pi/6)}=1/sqrt{2}=cos (pi/12)-sin (pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$. Fourthly, for $x=5pi/6>pi/2>pi/4$, we get what you are saying: $sqrt{1-sin (5pi/6)}=1/sqrt{2}=sin (5pi/12)-cos (5pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$.
            – farruhota
            2 days ago






            Firstly, $0le 1-sin xle 2<9$. Secondly, $sqrt{9}=3$ and $pm sqrt{9}=pm 3$. Thirdly, $0<x<pi/4$. For example, when $x=pi/6$, $sqrt{1-sin (pi/6)}=1/sqrt{2}=cos (pi/12)-sin (pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$. Fourthly, for $x=5pi/6>pi/2>pi/4$, we get what you are saying: $sqrt{1-sin (5pi/6)}=1/sqrt{2}=sin (5pi/12)-cos (5pi/12)=frac{1+sqrt{3}}{2sqrt{2}}-frac{sqrt{3}-1}{2sqrt{2}}$.
            – farruhota
            2 days ago




















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