If tan theta = x sin phi/ (1 -x cos phi) and tan phi = y sin theta / (1- y cos theta). then x/y =
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tanθ=xsinФ/(1-xcosФ)
or, cotθ=(1-xcosФ)/xsinФ
or, cotθ=(1/xsinФ)-cotФ
or, 1/xsinФ=cotθ+cotФ -----------------(1)
tanФ=ysinθ/(1-ycosθ)
or, cotФ=(1-ycosθ)/ysinθ
or, cotФ=(1/ysinθ)-cotθ
or, cotФ+cotθ=1/ysinθ
or, 1/xsinФ=1/ysinθ [using(1)]
or, xsinФ=ysinθ
or, x/y=sinθ/sinФ Ans.
or, cotθ=(1-xcosФ)/xsinФ
or, cotθ=(1/xsinФ)-cotФ
or, 1/xsinФ=cotθ+cotФ -----------------(1)
tanФ=ysinθ/(1-ycosθ)
or, cotФ=(1-ycosθ)/ysinθ
or, cotФ=(1/ysinθ)-cotθ
or, cotФ+cotθ=1/ysinθ
or, 1/xsinФ=1/ysinθ [using(1)]
or, xsinФ=ysinθ
or, x/y=sinθ/sinФ Ans.
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