If 5sinhx-coshx=5 find tanhx

Solution:

If $5 \sinh x-\cosh x=5$ so, find $\tanh x$

Given, $5 \sinh x-\cosh x=5$

Dividing by $cosh x$,

we get, $5 \tanh x-1=5 \operatorname{sech} x$

Squaring both sides,

we get $(5 \tanh x-1)^{2}=25 \operatorname{sech}^{2} x$

$= 25 \tanh ^{2} x+1-10 \tanh ^{2} x=25\left(1-\tanh ^{2} x\right)$

$= 50 \tanh ^{2} x-10 \operatorname{tanhx}-24=0$

$=25 \tanh ^{2} x-5 \tanh x-12=0$

$\tanh x=\frac{5 \pm \sqrt{25+1200}}{50}=\frac{5 \pm \sqrt{1225}}{50}$

$=\frac{5 \pm 35}{50}=\frac{40}{50}, \frac{-30}{50}=\frac{4}{5},-\frac{3}{5}$

So the values of $\tanh x$ are: $\frac{-3}{5} or \ \frac{4}{5}$