Question 1 - A-Level Maths

Use the derivatives of $\sin x$ and $\cos x$ to prove that the derivative of $\tan x$ is $\sec ^ { 2 } x$.
The function f is given by $\mathrm { f } : x \propto 2 + \frac { 3 } { x + 2 } , x \in \mathbb { R } , x \neq - 2$.
1. Express $2 + \frac { 3 } { x + 2 }$ as a single fraction.
2. Find an expression for $\mathrm { f } ^ { - 1 } ( x )$.
3. Write down the domain of $\mathrm { f } ^ { - 1 }$.
4. (a) Express as a fraction in its simplest form
$$\frac { 2 } { x - 3 } + \frac { 13 } { x ^ { 2 } + 4 x - 21 }$$
Hence solve $$\frac { 2 } { x - 3 } + \frac { 13 } { x ^ { 2 } + 4 x - 21 } = 1$$