IF #ptan(a)+qtan(b)=(p+q)tan{(a+b)/2}# then show that #{cos(a)/cos(b)}=p/q #?

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Answer 1 · 2018-06-18T03:35:08

#ptan(a)+qtan(b)=(p+q)tan{(a+b)/2}#

#=>ptan(a)+qtan(b)=(p+q)(2sin((a+b)/2)cos((a-b)/2))/(2cos((a+b)/2)cos((a-b)/2))#

#=>ptan(a)+qtan(b)=(p+q)(sin(a)+sin(b))/(cos(a)+cos(b))#

#=>(ptan(a)+qtan(b))(cos(a)+cos(b)) =(p+q)(sin(a)+sin(b))#

#=>ptan(a)cos(a)+qtan(b)cos(b)+qtan(b)cos(a)+ptan(a)cos(b) =p sin(a)+qsin(b)+qsin(a)+psin(b)#

#=>p sin(a)+q sin(b)+qtan(b)cos(a)+ptan(a)cos(b) =p sin(a)+qsin(b)+qsin(a)+p sin(b)#

#=>qtan(b)cos(a)+ptan(a)cos(b) =qsin(a)+p sin(b)#

#=>qtan(b)cos(a)-q sin (a)=p sin(b) -ptan(a)cos(b)#

#=>(q(sin(b)cos(a)-sin (a)cos(b)))/cos(b)=(p (sin(b)cos(a) -sin(a)cos(b)))/cos(a)#

#=>(q(sin(b-a)))/cos(b)=(p (sin(b-a)))/cos(a)#

#=>cos(a)/cos(b)=p/q#

Proved