Prove that $frac sin X+sin Y-sin(X+Y)sin X+sin Y+sin(X+Y)=tanfracX2tanfracY2$ [closed]

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Prove that $frac sin X+sin Y-sin(X+Y)sin X+sin Y+sin(X+Y)=tanfracX2tanfracY2$



I tried expanding $frac sin X+sin Y-sin(X+Y)sin X+sin Y+sin(X+Y)$ and expanding $tanfracX2tanfracY2$ and then cross multiplying , but I can't seem to find the answer.










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closed as off-topic by user21820, Xander Henderson, amWhy, alans, rschwieb Sep 13 at 16:11


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


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    Prove that $frac sin X+sin Y-sin(X+Y)sin X+sin Y+sin(X+Y)=tanfracX2tanfracY2$



    I tried expanding $frac sin X+sin Y-sin(X+Y)sin X+sin Y+sin(X+Y)$ and expanding $tanfracX2tanfracY2$ and then cross multiplying , but I can't seem to find the answer.










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    closed as off-topic by user21820, Xander Henderson, amWhy, alans, rschwieb Sep 13 at 16:11


    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." – user21820, Xander Henderson, amWhy, alans, rschwieb
    If this question can be reworded to fit the rules in the help center, please edit the question.














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      up vote
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      Prove that $frac sin X+sin Y-sin(X+Y)sin X+sin Y+sin(X+Y)=tanfracX2tanfracY2$



      I tried expanding $frac sin X+sin Y-sin(X+Y)sin X+sin Y+sin(X+Y)$ and expanding $tanfracX2tanfracY2$ and then cross multiplying , but I can't seem to find the answer.










      share|cite|improve this question















      Prove that $frac sin X+sin Y-sin(X+Y)sin X+sin Y+sin(X+Y)=tanfracX2tanfracY2$



      I tried expanding $frac sin X+sin Y-sin(X+Y)sin X+sin Y+sin(X+Y)$ and expanding $tanfracX2tanfracY2$ and then cross multiplying , but I can't seem to find the answer.







      algebra-precalculus trigonometry






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      edited Sep 13 at 14:59









      user21820

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      asked Sep 13 at 5:38









      mampuuu

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      closed as off-topic by user21820, Xander Henderson, amWhy, alans, rschwieb Sep 13 at 16:11


      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." – user21820, Xander Henderson, amWhy, alans, rschwieb
      If this question can be reworded to fit the rules in the help center, please edit the question.




      closed as off-topic by user21820, Xander Henderson, amWhy, alans, rschwieb Sep 13 at 16:11


      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." – user21820, Xander Henderson, amWhy, alans, rschwieb
      If this question can be reworded to fit the rules in the help center, please edit the question.




















          4 Answers
          4






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          Write $t=tan(x/2)$ and $u=tan(y/2)$.
          Then
          $$sin x=frac2t1+t^2,$$
          $$sin y=frac2u1+u^2$$
          and
          $$sin(x+y)=sin xcos y+cos xsin y=frac2t(1-u^2)+2u(1-t^2)(1+t^2)(1+u^2).$$
          Therefore
          $$sin x+sin ypmsin(x+y)
          =frac2t(1+u^2)+2u(1+t^2)pmleft(2t(1-u^2)+2u(1-t^2)right)(1+t^2)(1+u^2).$$
          Now expand out both possibilities.






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            up vote
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            Start with the LHS, use sum-to-product on sin(X)+sin(Y), and the double angle formula on sin(X+Y), the expression is equal to
            $$
            fracsin(fracX+Y2)cos(fracX-Y2)-sin(fracX+Y2)cos(fracX+Y2)sin(fracX+Y2)cos(fracX-Y2)+sin(fracX+Y2)cos(fracX+Y2)
            $$
            cancelling sin((X+Y)/2) and use sum-to-product again,
            $$
            =
            fracsinfracX2sinfracY2cosfracX2cosfracY2
            $$
            which is the RHS.






            share|cite|improve this answer




















            • mathworld.wolfram.com/ProsthaphaeresisFormulas.html
              – lab bhattacharjee
              Sep 13 at 5:57

















            up vote
            4
            down vote













            Denote: $x=a+b, y=a-b$, then:
            $$frac sin (a+b)+sin (a-b)-sin(2a)sin (a+b)+sin (a-b)+sin(2a)=\
            fracsin acos b+requirecancelcancelcos asin b+sin acos b-requirecancelcancelcos asin b-sin (2a)sin acos b+requirecancelcancelcos asin b+sin acos b-requirecancelcancelcos asin b+sin (2a)=\
            frac2sin acos b-2sin acos a2sin acos b+2sin acos a=\
            fraccos b-cos acos b+cos a=\
            frac-sin fracb-a2sin fracb+a2cos fracb-a2cos fracb+a2=\
            frac-sin (-frac y2)sin (frac x2)cos (-frac y2)cos (frac x2)=\
            tanfracX2tanfracY2.$$






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              beginalign*fracsin x+sin y-sin(x+y)sin x+sin y+sin (x+y)&=dfrac2sindfracx+y2cosdfracx-y2-2sindfracx+y2cos dfracx+y22sindfracx+y2cosdfracx-y2+2sindfracx+y2cos dfracx+y2\&=dfraccosdfracx-y2-cos dfracx+y2cosdfracx-y2+cos dfracx+y2\&=dfrac-2sindfracx2sindfrac-y22cosdfracx2cos dfrac-y2\&=tandfracx2tandfracy2.endalign*






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                4 Answers
                4






                active

                oldest

                votes








                4 Answers
                4






                active

                oldest

                votes









                active

                oldest

                votes






                active

                oldest

                votes








                up vote
                4
                down vote



                accepted










                Write $t=tan(x/2)$ and $u=tan(y/2)$.
                Then
                $$sin x=frac2t1+t^2,$$
                $$sin y=frac2u1+u^2$$
                and
                $$sin(x+y)=sin xcos y+cos xsin y=frac2t(1-u^2)+2u(1-t^2)(1+t^2)(1+u^2).$$
                Therefore
                $$sin x+sin ypmsin(x+y)
                =frac2t(1+u^2)+2u(1+t^2)pmleft(2t(1-u^2)+2u(1-t^2)right)(1+t^2)(1+u^2).$$
                Now expand out both possibilities.






                share|cite|improve this answer
























                  up vote
                  4
                  down vote



                  accepted










                  Write $t=tan(x/2)$ and $u=tan(y/2)$.
                  Then
                  $$sin x=frac2t1+t^2,$$
                  $$sin y=frac2u1+u^2$$
                  and
                  $$sin(x+y)=sin xcos y+cos xsin y=frac2t(1-u^2)+2u(1-t^2)(1+t^2)(1+u^2).$$
                  Therefore
                  $$sin x+sin ypmsin(x+y)
                  =frac2t(1+u^2)+2u(1+t^2)pmleft(2t(1-u^2)+2u(1-t^2)right)(1+t^2)(1+u^2).$$
                  Now expand out both possibilities.






                  share|cite|improve this answer






















                    up vote
                    4
                    down vote



                    accepted







                    up vote
                    4
                    down vote



                    accepted






                    Write $t=tan(x/2)$ and $u=tan(y/2)$.
                    Then
                    $$sin x=frac2t1+t^2,$$
                    $$sin y=frac2u1+u^2$$
                    and
                    $$sin(x+y)=sin xcos y+cos xsin y=frac2t(1-u^2)+2u(1-t^2)(1+t^2)(1+u^2).$$
                    Therefore
                    $$sin x+sin ypmsin(x+y)
                    =frac2t(1+u^2)+2u(1+t^2)pmleft(2t(1-u^2)+2u(1-t^2)right)(1+t^2)(1+u^2).$$
                    Now expand out both possibilities.






                    share|cite|improve this answer












                    Write $t=tan(x/2)$ and $u=tan(y/2)$.
                    Then
                    $$sin x=frac2t1+t^2,$$
                    $$sin y=frac2u1+u^2$$
                    and
                    $$sin(x+y)=sin xcos y+cos xsin y=frac2t(1-u^2)+2u(1-t^2)(1+t^2)(1+u^2).$$
                    Therefore
                    $$sin x+sin ypmsin(x+y)
                    =frac2t(1+u^2)+2u(1+t^2)pmleft(2t(1-u^2)+2u(1-t^2)right)(1+t^2)(1+u^2).$$
                    Now expand out both possibilities.







                    share|cite|improve this answer












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                    answered Sep 13 at 5:44









                    Lord Shark the Unknown

                    91k955117




                    91k955117




















                        up vote
                        6
                        down vote













                        Start with the LHS, use sum-to-product on sin(X)+sin(Y), and the double angle formula on sin(X+Y), the expression is equal to
                        $$
                        fracsin(fracX+Y2)cos(fracX-Y2)-sin(fracX+Y2)cos(fracX+Y2)sin(fracX+Y2)cos(fracX-Y2)+sin(fracX+Y2)cos(fracX+Y2)
                        $$
                        cancelling sin((X+Y)/2) and use sum-to-product again,
                        $$
                        =
                        fracsinfracX2sinfracY2cosfracX2cosfracY2
                        $$
                        which is the RHS.






                        share|cite|improve this answer




















                        • mathworld.wolfram.com/ProsthaphaeresisFormulas.html
                          – lab bhattacharjee
                          Sep 13 at 5:57














                        up vote
                        6
                        down vote













                        Start with the LHS, use sum-to-product on sin(X)+sin(Y), and the double angle formula on sin(X+Y), the expression is equal to
                        $$
                        fracsin(fracX+Y2)cos(fracX-Y2)-sin(fracX+Y2)cos(fracX+Y2)sin(fracX+Y2)cos(fracX-Y2)+sin(fracX+Y2)cos(fracX+Y2)
                        $$
                        cancelling sin((X+Y)/2) and use sum-to-product again,
                        $$
                        =
                        fracsinfracX2sinfracY2cosfracX2cosfracY2
                        $$
                        which is the RHS.






                        share|cite|improve this answer




















                        • mathworld.wolfram.com/ProsthaphaeresisFormulas.html
                          – lab bhattacharjee
                          Sep 13 at 5:57












                        up vote
                        6
                        down vote










                        up vote
                        6
                        down vote









                        Start with the LHS, use sum-to-product on sin(X)+sin(Y), and the double angle formula on sin(X+Y), the expression is equal to
                        $$
                        fracsin(fracX+Y2)cos(fracX-Y2)-sin(fracX+Y2)cos(fracX+Y2)sin(fracX+Y2)cos(fracX-Y2)+sin(fracX+Y2)cos(fracX+Y2)
                        $$
                        cancelling sin((X+Y)/2) and use sum-to-product again,
                        $$
                        =
                        fracsinfracX2sinfracY2cosfracX2cosfracY2
                        $$
                        which is the RHS.






                        share|cite|improve this answer












                        Start with the LHS, use sum-to-product on sin(X)+sin(Y), and the double angle formula on sin(X+Y), the expression is equal to
                        $$
                        fracsin(fracX+Y2)cos(fracX-Y2)-sin(fracX+Y2)cos(fracX+Y2)sin(fracX+Y2)cos(fracX-Y2)+sin(fracX+Y2)cos(fracX+Y2)
                        $$
                        cancelling sin((X+Y)/2) and use sum-to-product again,
                        $$
                        =
                        fracsinfracX2sinfracY2cosfracX2cosfracY2
                        $$
                        which is the RHS.







                        share|cite|improve this answer












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                        share|cite|improve this answer










                        answered Sep 13 at 5:46









                        user10354138

                        2,3626




                        2,3626











                        • mathworld.wolfram.com/ProsthaphaeresisFormulas.html
                          – lab bhattacharjee
                          Sep 13 at 5:57
















                        • mathworld.wolfram.com/ProsthaphaeresisFormulas.html
                          – lab bhattacharjee
                          Sep 13 at 5:57















                        mathworld.wolfram.com/ProsthaphaeresisFormulas.html
                        – lab bhattacharjee
                        Sep 13 at 5:57




                        mathworld.wolfram.com/ProsthaphaeresisFormulas.html
                        – lab bhattacharjee
                        Sep 13 at 5:57










                        up vote
                        4
                        down vote













                        Denote: $x=a+b, y=a-b$, then:
                        $$frac sin (a+b)+sin (a-b)-sin(2a)sin (a+b)+sin (a-b)+sin(2a)=\
                        fracsin acos b+requirecancelcancelcos asin b+sin acos b-requirecancelcancelcos asin b-sin (2a)sin acos b+requirecancelcancelcos asin b+sin acos b-requirecancelcancelcos asin b+sin (2a)=\
                        frac2sin acos b-2sin acos a2sin acos b+2sin acos a=\
                        fraccos b-cos acos b+cos a=\
                        frac-sin fracb-a2sin fracb+a2cos fracb-a2cos fracb+a2=\
                        frac-sin (-frac y2)sin (frac x2)cos (-frac y2)cos (frac x2)=\
                        tanfracX2tanfracY2.$$






                        share|cite|improve this answer
























                          up vote
                          4
                          down vote













                          Denote: $x=a+b, y=a-b$, then:
                          $$frac sin (a+b)+sin (a-b)-sin(2a)sin (a+b)+sin (a-b)+sin(2a)=\
                          fracsin acos b+requirecancelcancelcos asin b+sin acos b-requirecancelcancelcos asin b-sin (2a)sin acos b+requirecancelcancelcos asin b+sin acos b-requirecancelcancelcos asin b+sin (2a)=\
                          frac2sin acos b-2sin acos a2sin acos b+2sin acos a=\
                          fraccos b-cos acos b+cos a=\
                          frac-sin fracb-a2sin fracb+a2cos fracb-a2cos fracb+a2=\
                          frac-sin (-frac y2)sin (frac x2)cos (-frac y2)cos (frac x2)=\
                          tanfracX2tanfracY2.$$






                          share|cite|improve this answer






















                            up vote
                            4
                            down vote










                            up vote
                            4
                            down vote









                            Denote: $x=a+b, y=a-b$, then:
                            $$frac sin (a+b)+sin (a-b)-sin(2a)sin (a+b)+sin (a-b)+sin(2a)=\
                            fracsin acos b+requirecancelcancelcos asin b+sin acos b-requirecancelcancelcos asin b-sin (2a)sin acos b+requirecancelcancelcos asin b+sin acos b-requirecancelcancelcos asin b+sin (2a)=\
                            frac2sin acos b-2sin acos a2sin acos b+2sin acos a=\
                            fraccos b-cos acos b+cos a=\
                            frac-sin fracb-a2sin fracb+a2cos fracb-a2cos fracb+a2=\
                            frac-sin (-frac y2)sin (frac x2)cos (-frac y2)cos (frac x2)=\
                            tanfracX2tanfracY2.$$






                            share|cite|improve this answer












                            Denote: $x=a+b, y=a-b$, then:
                            $$frac sin (a+b)+sin (a-b)-sin(2a)sin (a+b)+sin (a-b)+sin(2a)=\
                            fracsin acos b+requirecancelcancelcos asin b+sin acos b-requirecancelcancelcos asin b-sin (2a)sin acos b+requirecancelcancelcos asin b+sin acos b-requirecancelcancelcos asin b+sin (2a)=\
                            frac2sin acos b-2sin acos a2sin acos b+2sin acos a=\
                            fraccos b-cos acos b+cos a=\
                            frac-sin fracb-a2sin fracb+a2cos fracb-a2cos fracb+a2=\
                            frac-sin (-frac y2)sin (frac x2)cos (-frac y2)cos (frac x2)=\
                            tanfracX2tanfracY2.$$







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                            answered Sep 13 at 8:06









                            farruhota

                            15.9k2734




                            15.9k2734




















                                up vote
                                1
                                down vote













                                beginalign*fracsin x+sin y-sin(x+y)sin x+sin y+sin (x+y)&=dfrac2sindfracx+y2cosdfracx-y2-2sindfracx+y2cos dfracx+y22sindfracx+y2cosdfracx-y2+2sindfracx+y2cos dfracx+y2\&=dfraccosdfracx-y2-cos dfracx+y2cosdfracx-y2+cos dfracx+y2\&=dfrac-2sindfracx2sindfrac-y22cosdfracx2cos dfrac-y2\&=tandfracx2tandfracy2.endalign*






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                                  up vote
                                  1
                                  down vote













                                  beginalign*fracsin x+sin y-sin(x+y)sin x+sin y+sin (x+y)&=dfrac2sindfracx+y2cosdfracx-y2-2sindfracx+y2cos dfracx+y22sindfracx+y2cosdfracx-y2+2sindfracx+y2cos dfracx+y2\&=dfraccosdfracx-y2-cos dfracx+y2cosdfracx-y2+cos dfracx+y2\&=dfrac-2sindfracx2sindfrac-y22cosdfracx2cos dfrac-y2\&=tandfracx2tandfracy2.endalign*






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                                    up vote
                                    1
                                    down vote










                                    up vote
                                    1
                                    down vote









                                    beginalign*fracsin x+sin y-sin(x+y)sin x+sin y+sin (x+y)&=dfrac2sindfracx+y2cosdfracx-y2-2sindfracx+y2cos dfracx+y22sindfracx+y2cosdfracx-y2+2sindfracx+y2cos dfracx+y2\&=dfraccosdfracx-y2-cos dfracx+y2cosdfracx-y2+cos dfracx+y2\&=dfrac-2sindfracx2sindfrac-y22cosdfracx2cos dfrac-y2\&=tandfracx2tandfracy2.endalign*






                                    share|cite|improve this answer












                                    beginalign*fracsin x+sin y-sin(x+y)sin x+sin y+sin (x+y)&=dfrac2sindfracx+y2cosdfracx-y2-2sindfracx+y2cos dfracx+y22sindfracx+y2cosdfracx-y2+2sindfracx+y2cos dfracx+y2\&=dfraccosdfracx-y2-cos dfracx+y2cosdfracx-y2+cos dfracx+y2\&=dfrac-2sindfracx2sindfrac-y22cosdfracx2cos dfrac-y2\&=tandfracx2tandfracy2.endalign*







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                                    answered Sep 13 at 6:36









                                    mengdie1982

                                    3,817216




                                    3,817216












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