Solve: tan(x).tan(y)=a ,x+y=2b (x,y)=?

Question

if tan(x).tan(y)=a & x+y=2b ,
then what are the values of x & y ?

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Answer 1 · 2017-05-10T19:08:05

#x=y=b = pmarctan(sqrta)#

For symmetry #x=y=b# so

#tan^2b=a# so #tan b = pmsqrt a#

and

#b = pmarctan(sqrta)#

Answer 2 · 2017-05-11T00:21:15

#tanx*tany=a.......[1]#

#=>(2sinxsiny)/(2cosxcosy)=a#

#=>(cos(x-y)-cos(x+y))/(cos(x-y)+cos(x+y))=a/1#

#=>(cos(x-y)+cos(x+y))/(cos(x-y)-cos(x+y))=1/a#

#=>(2cos(x-y))/(2cos(x+y))=(1+a)/(1-a)# [by componendo dividendo]

#=>(cos(x-y))/(cos(x+y))=(1+a)/(1-a)#

Given #x+y=2b......[2]#

So

#=>(cos(x-y))/(cos2b)=(1+a)/(1-a)#

#=>cos(x-y)=(1+a)/(1-a)xxcos2b#

#=>x-y=cos^-1((1+a)/(1-a)xxcos2b)........[3]#

Adding [2} and {3] we get

#x=b+1/2cos^-1((1+a)/(1-a)xxcos2b)#

Adding [3} from [2] we get

#y=b-1/2cos^-1((1+a)/(1-a)xxcos2b)#