Given
#sina/sinb=p and cosa/cosb=q#
We have
#sina=p*sinb ........[1]#
and
# cosa=q*cosb.....[2]#
So
#sin^2a+cos^2a=1#
#=>p^2sin^2b+q^2cos^2b=1#
#=>p^2tan^2b+q^2=sec^2b#
#=>p^2tan^2b+q^2=1+tan^2b#
#=>(p^2-1)tan^2b=(1-q^2)#
#=>tan^2b=(1-q^2)/(p^2-1)#
#=>tanb=pmsqrt((1-q^2)/(p^2-1))......[3]#
Similarly we can proceed for #tana#
Given
#sina/sinb=p and cosa/cosb=q#
We have
#sinb=sina/pand cosb=cosa/q#
So
#sin^2b+cos^2b=1#
#=>sin^2a/p^2+cos^2a/q^2=sin^2a+cos^2a#
#=>sin^2a(1/p^2-1)=cos^2a(1-1/q^2)#
#=>sin^2a/cos^2a=(1-1/q^2)/(1/p^2-1)#
#=>tan^2a=(1-1/q^2)/(1/p^2-1)#
#=>tana=pmsqrt((1-1/q^2)/(1/p^2-1))=pmp/qsqrt((q^2-1)/(1-p^2))#
#=>tana=pmp/qsqrt((1-q^2)/(p^2-1))#
Otherwise
Dividing [1] by [2]we have
#tana=p/qtanb#
using {3] we get
#tana=pmp/qsqrt((1-q^2)/(p^2-1))#