Prove that Tanh (x-y)= tan hx - tan hy/ 1-tan hx tan hy.

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Solution

$\tanh (x - y) = \frac{\sinh (x - y)}{\cosh (x - y)} = \frac{\sinh x \cosh y - \cosh x \sinh y}{\cosh x \cosh y - \sinh x \sinh y} Dividing numerator and denominator by \cosh x \cosh y = \frac{\frac{\sinh x \cosh y}{\cosh x \cosh y} - \frac{\cosh x \sinh y}{\cosh x \cosh y}}{\frac{\cosh x \cosh y}{\cosh x \cosh y} - \frac{\sinh x \sinh y}{\cosh x \cosh y}} = \frac{\tanh x - \tanh y}{1 - \tanh x \tanh y} = RHS Hence Proved .$

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