prove sinA+sinB/sinA-sinB =a+b/a-b.
Answers
Answer:
Q )SinA +SinB =a Cos A +CosB =b then prove that
Sin(A+B)= 2ab/ a^2+b^2
Sin(A+B) = sin A cos B +CosA SinB
Cos(A+B) = cos A cosB + SinA SinB
Finding a^2 +b^2 (1)
a^2 +b^2= sin^2A +sin^2B +2sinA SinB + Cos^2A +Cos^2B + 2Cos A Cos B
= 1+1 +2 ( CosA CosB + SinA SinB )
=2+2( CosA cos B +sinA SinB)
a^2+b^2 --2 =2 Cos(A-B) )
Cos (A-B) = (a^2+b^2 )/2 --1 (1)
Finding values of ab (2)
2ab =2 (sinA +SinB ) (CosA +CosB)
= 2 [ sinA CosA + SinA cos B + SinB Cos A + SinB Cos B]
2ab= 2 (1+ Cos A CosB + SinA SinB )
ab= [1+ cos(A --B)]
ab=1 +(a^2+b^2)/2 --1 from (1)
ab=( a^2+b^2)/2
Hence sin(A+B) = 2ab /a^2+b^2
LH S =R H S
Formulae used ::
1 )sin^2A + cos^2A = 1
2 )Cos ( A -B) = CosA Cos B + SinA SinB
Problem solving skills::
1 0bservation of relation ship
2 Finding a^2 + b^2
3 Finding answer
4 Recollecting formula and previous steps
Formulae 1 )sin^2A +Cos^2A =1
Cos (A +B)
Time for solving 20 to 25 minutes
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