How do I prove that $2 sin ((C+D)/2) cos ((C-D)/2) = sin C+sin D$ ?

Question

How do I prove that $2 sin ((C+D)/2) cos ((C-D)/2) = sin C+sin D$ ?

2 Answers

Impact of this question

42824 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-11-20T14:00:26

$"see explanation"$

Explanation:

$"using the "color(blue)"trigonometric identities"$

$•color(white)(x)sin(A+B)=sinAcosB+cosAsinB$

$•color(white)(x)sin(A-B)=sinAcosB-cosAsinB$

$"Adding the 2 equations gives"$

$sin(A+B)+sin(A-B)=2sinAcosB$

$"Subtracting the 2 equations gives"$

$sin(A+B)-sin(A-B)=2cosAsinB$

$"let "C=A+B" and "D=A-B$

$rArrA=(C+D)/2" and "B=(C-D)/2$

$rArrsinC+sinD=2sin((C+D)/2)cos((C-D)/2)$

Answer 2 · 2017-11-20T14:13:27

See the proof below

Explanation:

We need

$sin(a+b)=sinacosb+sinbcosa$

$cos(a-b)=cosacosb+sinasinb$

$sin^2a+cos^2a=1$

$sin2a=2sinacosa$

Therefore,

$LHS=2sin((C+D)/2)cos((C-D)/2)$

$=2(sin(C/2)cos(D/2)+sin(D/2)cos(C/2))(cos(C/2)cos(D/2)+sin(C/2)sin(D/2))$

$=2(sin(C/2)cos(C/2)cos^2(D/2)+sin(D/2)cos(D/2)sin^2(C/2)+sin(D/2)cos(D/2)cos^2(C/2)+sin(C/2)cos(C/2)sin^2(D/2))$

$=2*1/2(sinCcos^2(D/2)+sinDsin^2(C/2)+sinDcos^2(C/2)+sinCsin^2(D/2))$

$=sinC(cos^2(D/2)+sin^2(D/2))+sinD(sin^2(C/2)+cos^2(C/2))$

$=sinC+sinD$

$=RHS$

$QED$

Science

Math

Humanities

... and beyond

How do I prove that $2 sin ((C+D)/2) cos ((C-D)/2) = sin C+sin D$ ?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do I prove that 2 sin ((C+D)/2) cos ((C-D)/2) = sin C+sin D2 sin ((C+D)/2) cos ((C-D)/2) = sin C+sin D?

2 Answers

Explanation:

Explanation:

Related questions

Impact of this question

How do I prove that $2 sin ((C+D)/2) cos ((C-D)/2) = sin C+sin D$ ?