Question

If `3coshx+sinhx=3` find `cothx`?

Answer 1

`cothx=-5/3`

Explanation:

As `3coshx+sinhx=3`

`(3(e^x+e^(-x)))/2+(e^x-e^(-x))/2=3`

or `(4e^x+2e^(-x))/2=3`

or `2e^x+e^(-x)=3` and multilying each term by `e^x`

`2(e^x)^2-3e^x+1=0`

or `(2e^x-1)(e^x-1)=0`

i.e. `e^x=1` or `e^x=1/2`

As `cothx=(e^x+e^(-x))/(e^x-e^(-x))`

observe that if `e^x=1` i.e. `x=0`, `cothx` is not defined

hence `cothx=(1/2+2)/(1/2-2)=-(5/2)/(3/2)=-5/3`