If sinx+siny=a and cosx+cosy=b then find tan(x-y/2)

Tabansi 2020-10-18 Answered
If sinx+siny=aandcosx+cosy=b then find tan(xy2)
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Expert Answer

Benedict
Answered 2020-10-19 Author has 108 answers

sin(A+B)=sinAcosB+cosAsinBandsin(AB)=sinAcosBcosAsinB.
cos(A+B)=cosAcosBsinAsinBandcos(AB)=cosAcosB+sinAsinB.
These are true for all A and B. So, sin(A+B)+sin(AB)=2sinAcosBandcos(A+B)+cos(AB)=2cosAcosB.
Therefore relating A and B to x and y we getA+B=xandAB=y.FromtheseA=x+y2andB=xy2.
We can now write sinx+siny=2sin12(x+y)cos12(xy)=aandcosx+cosy=2cos12(x+y)cos12(xy)=b.
Therefore, dividing these two we get tan12(x+y)=ab.
If we square each of the original equations we get:
sin2x+sin2y+2sinx.siny=a2andcos2x+cos2y+2cosx.cosy=b2
Adding these two equations we get 2+2cos(xy)=a2+b2[sin2+cos2=1 for h xandy]
We can expand cos(xy) into 2cos2((xy)/2)1, so we can write

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Jeffrey Jordon
Answered 2021-09-16 Author has 2495 answers

The value of tan(xy2)

Step-by-step explanation:

Given: sinx+siny=a,cosx+cosy=b

To find: The value of tan(xy2)

Solution:

sinx+siny=a

We know, sinx+siny=2sin(x+y2)cos(xy2)

2sin(x+y2)cos(xy2)=a....(1)

cosx+cosy=b

We know, cosx+cosy=2cos(x+y2)sin(xy2)

2cos(x+y2)sin(xy2)=b...(2)

Divide (1) and (2),

2sin(x+y2)cos(xy2)2cos(x+y2)sin(xy2)

Therefore, the value of tan(x+y2)=ab

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