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1)sinx+siny=a,cosx+cosy=b findsin(x+y)&cos2x+cos2y asin 2 x+bcos 2 x=m,bsin 2 y+acos 2 y=n&atanx=btany then prove that 1/n+1/m=1/a=1/b

1)sinx+siny=a,cosx+cosy=b
findsin(x+y)&cos2x+cos2y
  • asin2x+bcos2x=m,bsin2y+acos2y=n&atanx=btany then prove that  1/n+1/m=1/a=1/b 

Grade:11

2 Answers

sarthak
22 Points
9 years ago
it is1/a+1/b
rajesh
30 Points
9 years ago
asin2x+bcos2x=m      1
bsin2x+acos2x=n       2
in 1 and 2 take cos2x common and send it to rhs in both equations then the resulting equations would be
atan2x+b=msec2x
=>atan2x+b=m+mtan2x
=>tan2x=m-b/a-m      3
similarly second equation would be 
tan2y=n-a/b-n            4
given atanx=btany
squaring on both sides and regrouping we get
tan2x/tan2y=b2/a2        5
divide 3 and 4 and equte it to 5
we get 
(m-b)(n-b)/(n-a)(m-a)=b2/a2
by cross multiplication and regrouping of the above equation we get 
1/m+1/n=1/a+1/b
 
 
tan((x+y)/2)=sinx+siny/cosx+cosy
                 =a/b
sin(x+y)=2tan((x+y)/2)/1+tan2((x+y)/2)
substituting the abve value we get 
sin(x+y)=2a/1+b2

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