Let's recall the definitions: `tanh(x) = (sinh(x))/(cosh(x)),` `se ch(x) = 1/(cosh(x)).` Also, `cosh(x) = (e^x + e^(-x))/2` and `sinh(x) = (e^x - e^(-x))/2.`

Now the left side of our identity may be rewritten as

`tanh^2(x) + sec h^2(x) = (sinh^2(x) + 1)/(cosh^2(x)).`

While it is well-known that `sinh^2(x) + 1 = cosh^2(x),` we can prove this directly:

`sinh^2(x) + 1 = ((e^x - e^(-x))/2)^2 + 1 = ((e^x)^2 - 2 + (e^(-x))^2 + 4)/4 =`

`=((e^x)^2 + 2 + (e^(-x))^2)/4 =((e^x + e^(-x))/2)^2 = cosh^2(x).`

This way the left side is equal to `1,` which is the right side. This way the identity is proved.

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