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Prove the tangent rule: Napier's analogy...

Prove the tangent rule: Napier's analogy `tan((A-B)/2)=(a-b)/(a+b) cot(C/2)`

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`R.H.S. = (a-b)/(a+b)cot(C/2)`
From sine rule,
`a = 2RsinA and b = 2RsinA`
Also, `A+B+C = pi => C = pi - (A+B)`
`=> (a-b)/(a+b)cot(C/2) = (2RsinA-2RsinB)/(2RsinA+2RsinB) cot ((pi-(A+B))/2)`
`=(2R(sinA-sinB))/(2R(sinA+sinB)) cot (pi/2 - (A+B)/2)`
`=(sinA-sinB)/(sinA+sinB) tan ((A+B)/2)`
`=(2cos((A+B)/2)sin((A-B)/2))/(2sin((A+B)/2)cos((A-B)/2))sin((A+B)/2)/cos((A+B)/2)`
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