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## Homework Statement

After having struggled yesterday with this as much as I could, I am posting this problem here-

if ##ax+bsec(tan^{-1}x)=c## and ##ay+bsec(tan^{-1}y)=c##, then prove that ##\frac{x+y}{1-xy}=\frac{2ac}{a^2-c^2}##.

## Homework Equations

## The Attempt at a Solution

My attempt- Comparing both the equations, ##x=y## clearly(it is a dummy variable, sort of). So we basically need to find ##\frac{2x}{1-x^2}##. Put ##x=tan\theta##; so the given equation becomes-

##a tan\theta +b sec\theta=c##. From here, I don't know what to do. I tried to put it in the form of ##a sin\theta -(c) cos\theta=-b##, then divide and multiply it with ##\sqrt{(a^2+c^2)}## and put it in the auxiliary form(using ##a=rcos\theta,c=rsin\theta##); but alas; no help. By just working backwards, we see that the transformations ##a=rcos\theta## and ##c=rsin\theta## give the answer; and these are also the transforms I need to get my auxiliary form I mentioned above. But how do I connect them beats me. and Please help; I'd really appreciate it. Thanks in advance!