Express $x$ in terms of $a$ and $b$: $sin^-1 frac2a1+a^2 + sin^-1frac2b1+b^2 = 2tan^-1x$

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Find the value of $x$ from the following equation in terms of $a$ and $b$
$$sin^-1 2aover1+a^2 + sin^-12bover1+b^2 = 2tan^-1x$$




I tried to expand the LHS using the formula $$sin^-1c+sin^-1d = sin^-1left(csqrt1+d^2 + dsqrt1+c^2right)$$



But it didn't work out. Could anyone help me out?










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  • I think you left out the parentheses on the right-hand side of the last formula. Please check that you agree with my edit.
    – saulspatz
    Sep 8 at 13:00










  • @saulspatz Oh yes. Thanks mate
    – S.Nep
    Sep 8 at 13:14














up vote
1
down vote

favorite













Find the value of $x$ from the following equation in terms of $a$ and $b$
$$sin^-1 2aover1+a^2 + sin^-12bover1+b^2 = 2tan^-1x$$




I tried to expand the LHS using the formula $$sin^-1c+sin^-1d = sin^-1left(csqrt1+d^2 + dsqrt1+c^2right)$$



But it didn't work out. Could anyone help me out?










share|cite|improve this question























  • I think you left out the parentheses on the right-hand side of the last formula. Please check that you agree with my edit.
    – saulspatz
    Sep 8 at 13:00










  • @saulspatz Oh yes. Thanks mate
    – S.Nep
    Sep 8 at 13:14












up vote
1
down vote

favorite









up vote
1
down vote

favorite












Find the value of $x$ from the following equation in terms of $a$ and $b$
$$sin^-1 2aover1+a^2 + sin^-12bover1+b^2 = 2tan^-1x$$




I tried to expand the LHS using the formula $$sin^-1c+sin^-1d = sin^-1left(csqrt1+d^2 + dsqrt1+c^2right)$$



But it didn't work out. Could anyone help me out?










share|cite|improve this question
















Find the value of $x$ from the following equation in terms of $a$ and $b$
$$sin^-1 2aover1+a^2 + sin^-12bover1+b^2 = 2tan^-1x$$




I tried to expand the LHS using the formula $$sin^-1c+sin^-1d = sin^-1left(csqrt1+d^2 + dsqrt1+c^2right)$$



But it didn't work out. Could anyone help me out?







trigonometry






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edited Sep 8 at 17:54









Blue

44.4k868142




44.4k868142










asked Sep 8 at 12:50









S.Nep

394




394











  • I think you left out the parentheses on the right-hand side of the last formula. Please check that you agree with my edit.
    – saulspatz
    Sep 8 at 13:00










  • @saulspatz Oh yes. Thanks mate
    – S.Nep
    Sep 8 at 13:14
















  • I think you left out the parentheses on the right-hand side of the last formula. Please check that you agree with my edit.
    – saulspatz
    Sep 8 at 13:00










  • @saulspatz Oh yes. Thanks mate
    – S.Nep
    Sep 8 at 13:14















I think you left out the parentheses on the right-hand side of the last formula. Please check that you agree with my edit.
– saulspatz
Sep 8 at 13:00




I think you left out the parentheses on the right-hand side of the last formula. Please check that you agree with my edit.
– saulspatz
Sep 8 at 13:00












@saulspatz Oh yes. Thanks mate
– S.Nep
Sep 8 at 13:14




@saulspatz Oh yes. Thanks mate
– S.Nep
Sep 8 at 13:14










2 Answers
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Divide both sides by $2$, then using the $arctan(x)=y implies x=tan(y)$ you will get that $$x=tanleft(fracarcsinleft(frac2a1+a^2right)+arcsinleft(frac2b1+b^2right)2right)$$






share|cite|improve this answer






















  • Welcome to MSE. Your formatting will look a lot nicer if you use arctan and tan instead of tan and arctan.
    – saulspatz
    Sep 8 at 13:02










  • @saulspatz thank you!
    – Donna Caligine
    Sep 8 at 13:03










  • My pleasure. I also suggest that you put all mathematical expressions, including numbers, inside dollars signs. $2$ comes out as $2$.
    – saulspatz
    Sep 8 at 13:09










  • Welcome! Another typesetting hint: use left( ... right) instead of just ( ... ) for parentheses that need to contain large expressions; they will grow as needed.
    – Théophile
    Sep 8 at 13:11










  • Now I'm one of you guys hahaha, thank you again for the great suggestions!
    – Donna Caligine
    Sep 8 at 13:16

















up vote
0
down vote













Let $arctan a=u,a=tan u$



If $-dfracpi2le2uledfracpi2iff -1le ale1$ $$P=arcsindfrac2a1+a^2=2arctan a$$



If $2u>dfracpi2,P=pi-2arctan a$



If $2u<-dfracpi2,P=-pi-2arctan a$



Now use my answer in showing $arctan(frac23) = frac12 arctan(frac125)$






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    2 Answers
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    2 Answers
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    Divide both sides by $2$, then using the $arctan(x)=y implies x=tan(y)$ you will get that $$x=tanleft(fracarcsinleft(frac2a1+a^2right)+arcsinleft(frac2b1+b^2right)2right)$$






    share|cite|improve this answer






















    • Welcome to MSE. Your formatting will look a lot nicer if you use arctan and tan instead of tan and arctan.
      – saulspatz
      Sep 8 at 13:02










    • @saulspatz thank you!
      – Donna Caligine
      Sep 8 at 13:03










    • My pleasure. I also suggest that you put all mathematical expressions, including numbers, inside dollars signs. $2$ comes out as $2$.
      – saulspatz
      Sep 8 at 13:09










    • Welcome! Another typesetting hint: use left( ... right) instead of just ( ... ) for parentheses that need to contain large expressions; they will grow as needed.
      – Théophile
      Sep 8 at 13:11










    • Now I'm one of you guys hahaha, thank you again for the great suggestions!
      – Donna Caligine
      Sep 8 at 13:16














    up vote
    3
    down vote













    Divide both sides by $2$, then using the $arctan(x)=y implies x=tan(y)$ you will get that $$x=tanleft(fracarcsinleft(frac2a1+a^2right)+arcsinleft(frac2b1+b^2right)2right)$$






    share|cite|improve this answer






















    • Welcome to MSE. Your formatting will look a lot nicer if you use arctan and tan instead of tan and arctan.
      – saulspatz
      Sep 8 at 13:02










    • @saulspatz thank you!
      – Donna Caligine
      Sep 8 at 13:03










    • My pleasure. I also suggest that you put all mathematical expressions, including numbers, inside dollars signs. $2$ comes out as $2$.
      – saulspatz
      Sep 8 at 13:09










    • Welcome! Another typesetting hint: use left( ... right) instead of just ( ... ) for parentheses that need to contain large expressions; they will grow as needed.
      – Théophile
      Sep 8 at 13:11










    • Now I'm one of you guys hahaha, thank you again for the great suggestions!
      – Donna Caligine
      Sep 8 at 13:16












    up vote
    3
    down vote










    up vote
    3
    down vote









    Divide both sides by $2$, then using the $arctan(x)=y implies x=tan(y)$ you will get that $$x=tanleft(fracarcsinleft(frac2a1+a^2right)+arcsinleft(frac2b1+b^2right)2right)$$






    share|cite|improve this answer














    Divide both sides by $2$, then using the $arctan(x)=y implies x=tan(y)$ you will get that $$x=tanleft(fracarcsinleft(frac2a1+a^2right)+arcsinleft(frac2b1+b^2right)2right)$$







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited Sep 8 at 13:13









    Larry

    31814




    31814










    answered Sep 8 at 12:57









    Donna Caligine

    356




    356











    • Welcome to MSE. Your formatting will look a lot nicer if you use arctan and tan instead of tan and arctan.
      – saulspatz
      Sep 8 at 13:02










    • @saulspatz thank you!
      – Donna Caligine
      Sep 8 at 13:03










    • My pleasure. I also suggest that you put all mathematical expressions, including numbers, inside dollars signs. $2$ comes out as $2$.
      – saulspatz
      Sep 8 at 13:09










    • Welcome! Another typesetting hint: use left( ... right) instead of just ( ... ) for parentheses that need to contain large expressions; they will grow as needed.
      – Théophile
      Sep 8 at 13:11










    • Now I'm one of you guys hahaha, thank you again for the great suggestions!
      – Donna Caligine
      Sep 8 at 13:16
















    • Welcome to MSE. Your formatting will look a lot nicer if you use arctan and tan instead of tan and arctan.
      – saulspatz
      Sep 8 at 13:02










    • @saulspatz thank you!
      – Donna Caligine
      Sep 8 at 13:03










    • My pleasure. I also suggest that you put all mathematical expressions, including numbers, inside dollars signs. $2$ comes out as $2$.
      – saulspatz
      Sep 8 at 13:09










    • Welcome! Another typesetting hint: use left( ... right) instead of just ( ... ) for parentheses that need to contain large expressions; they will grow as needed.
      – Théophile
      Sep 8 at 13:11










    • Now I'm one of you guys hahaha, thank you again for the great suggestions!
      – Donna Caligine
      Sep 8 at 13:16















    Welcome to MSE. Your formatting will look a lot nicer if you use arctan and tan instead of tan and arctan.
    – saulspatz
    Sep 8 at 13:02




    Welcome to MSE. Your formatting will look a lot nicer if you use arctan and tan instead of tan and arctan.
    – saulspatz
    Sep 8 at 13:02












    @saulspatz thank you!
    – Donna Caligine
    Sep 8 at 13:03




    @saulspatz thank you!
    – Donna Caligine
    Sep 8 at 13:03












    My pleasure. I also suggest that you put all mathematical expressions, including numbers, inside dollars signs. $2$ comes out as $2$.
    – saulspatz
    Sep 8 at 13:09




    My pleasure. I also suggest that you put all mathematical expressions, including numbers, inside dollars signs. $2$ comes out as $2$.
    – saulspatz
    Sep 8 at 13:09












    Welcome! Another typesetting hint: use left( ... right) instead of just ( ... ) for parentheses that need to contain large expressions; they will grow as needed.
    – Théophile
    Sep 8 at 13:11




    Welcome! Another typesetting hint: use left( ... right) instead of just ( ... ) for parentheses that need to contain large expressions; they will grow as needed.
    – Théophile
    Sep 8 at 13:11












    Now I'm one of you guys hahaha, thank you again for the great suggestions!
    – Donna Caligine
    Sep 8 at 13:16




    Now I'm one of you guys hahaha, thank you again for the great suggestions!
    – Donna Caligine
    Sep 8 at 13:16










    up vote
    0
    down vote













    Let $arctan a=u,a=tan u$



    If $-dfracpi2le2uledfracpi2iff -1le ale1$ $$P=arcsindfrac2a1+a^2=2arctan a$$



    If $2u>dfracpi2,P=pi-2arctan a$



    If $2u<-dfracpi2,P=-pi-2arctan a$



    Now use my answer in showing $arctan(frac23) = frac12 arctan(frac125)$






    share|cite|improve this answer
























      up vote
      0
      down vote













      Let $arctan a=u,a=tan u$



      If $-dfracpi2le2uledfracpi2iff -1le ale1$ $$P=arcsindfrac2a1+a^2=2arctan a$$



      If $2u>dfracpi2,P=pi-2arctan a$



      If $2u<-dfracpi2,P=-pi-2arctan a$



      Now use my answer in showing $arctan(frac23) = frac12 arctan(frac125)$






      share|cite|improve this answer






















        up vote
        0
        down vote










        up vote
        0
        down vote









        Let $arctan a=u,a=tan u$



        If $-dfracpi2le2uledfracpi2iff -1le ale1$ $$P=arcsindfrac2a1+a^2=2arctan a$$



        If $2u>dfracpi2,P=pi-2arctan a$



        If $2u<-dfracpi2,P=-pi-2arctan a$



        Now use my answer in showing $arctan(frac23) = frac12 arctan(frac125)$






        share|cite|improve this answer












        Let $arctan a=u,a=tan u$



        If $-dfracpi2le2uledfracpi2iff -1le ale1$ $$P=arcsindfrac2a1+a^2=2arctan a$$



        If $2u>dfracpi2,P=pi-2arctan a$



        If $2u<-dfracpi2,P=-pi-2arctan a$



        Now use my answer in showing $arctan(frac23) = frac12 arctan(frac125)$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Sep 8 at 13:59









        lab bhattacharjee

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