If sinx+siny=a,cosx+cosy=b, prove t...
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If ( sin x+sin y=a, cos x+cos y=b, ) prove that ( sin (x+y)=frac{2 a b}{a^{2}+b^{2}} )

JEE/Engineering Exams
Maths
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If sinx+siny=a,cosx+cosy=b, prove that sin(x+y)=2aba2+b2
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( sin x+sin y=a Rightarrow(sin x+sin y)^{2}=a^{2} )
( Rightarrow quad sin ^{2} x+sin ^{2} y+2 sin x sin y=a^{2} )
( 2+2(sin x sin y+cos x cos y)=a^{2}+b^{2} )
( Rightarrow quad 2+2 cos (x-y)=a^{2}+b^{2} )
( Rightarrow quad cos (x-y)=frac{a^{2}+b^{2}-2}{2} )
Vow, ( a b=sin x cos x+ ) sing cosy ( +sin x cos y+cos x sin y )
( Rightarrow quad 2 a b=sin 2 x+sin 2 y+2 operatorname{sen}(x+y) )
( Rightarrow quad 2 a b=2 sin (x+y) cos (x-y)+2 sin (x+y) )
( Rightarrow quad a b=sin (x+y)(cos (x-y)+1) quadleft(text { from } e q^{n} cdot 3right) )
( Rightarrow quad a b=sin (x+y)left(frac{a^{2}+b^{2}}{2}right) )
( left[sin (x+y)=frac{2 a b}{a^{2}+b^{2}}right] ) heree, proved

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