The given equation of a pair of straight lines is
#x^2(tan^2theta+cos^2theta)-2xytantheta+y^2sin^2theta=0#
#or,y^2-(2tantheta)/sin^2thetaxy+(tan^2theta+cos^2theta)/sin^2thetax^2=0#
Let the equations of component straight lines are #y-m_1x=0andy-m_2x=0#, where #m_1andm_2# are the slopes of the two lines.
So we can write
#(y-m_1x)(y-m_2x)=y^2-(2tantheta)/sin^2thetaxy+(tan^2theta+cos^2theta)/sin^2thetax^2#
#=>y^2-(m_1+m_2)xy+m_1m_2x^2=y^2-(2tantheta)/sin^2thetaxy+(tan^2theta+cos^2theta)/sin^2thetax^2#
Comparing both sides we get
#m_1+m_2=(2tantheta)/sin^2theta=(2sectheta)/ sintheta#
And
#m_1m_2=(tan^2theta+cos^2theta)/sin^2theta#
So
#(m_2-m_2)^2=(m_1+m_2)^2-4m_1m_2#
#=((2sectheta)/ sintheta)^2-(4(tan^2theta+cos^2theta))/sin^2theta#
#=4[(sec^2theta-tan^2theta-cos^2theta)/sin^2theta]#
#=4[(1-cos^2theta)/sin^2theta]#
#=4[(sin^2theta)/sin^2theta]#
#=4#
Hence #abs(m_1-m_2)=2#
So it proves that the difference of tangents of the angles made by the lines with the X-axis is 2
