Question 12 - A-Level Maths

Starting from the identities for $\sinh 2 x$ and $\cosh 2 x$, prove the identity $$\tanh 2 x = \frac { 2 \tanh x } { 1 + \tanh ^ { 2 } x }$$ 12
1. The function f is defined by $$\mathrm { f } ( x ) = \tanh x \quad ( x > 0 )$$ State the range of f
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(ii) Use part (a) and part (b)(i) to prove that $\tanh 2 x > \tanh x$ if $x > 0$