5.(a)The box below shows a student's attempt to prove the following identity for $a > b > 0$

$$\arctan a - \arctan b \equiv \arctan \frac { a - b } { 1 + a b }$$

Let $x = \arctan a$ and $y = \arctan b$ ,so that $a = \tan x$ and $b = \tan y$

$$\begin{aligned}
\text { So } \tan ( \arctan a - \arctan b ) & \equiv \tan ( x - y ) \\
& \equiv \frac { \tan x - \tan y } { 1 - \tan ^ { 2 } ( x y ) } \\
& \equiv \frac { a - b } { 1 - ( a b ) ^ { 2 } } \\
& \equiv \frac { a - a b + a b - b } { ( 1 - a b ) ( 1 + a b ) } \\
& \equiv \frac { a ( 1 - a b ) - b ( 1 - a b ) } { ( 1 - a b ) ( 1 + a b ) } \\
& \equiv \frac { a - b } { 1 + a b }
\end{aligned}$$

Taking arctan of both sides gives $\arctan a - \arctan b \equiv \arctan \frac { a - b } { 1 + a b }$ as required.

There are three errors in the proof where the working does not follow from the previous line.
\begin{enumerate}[label=(\roman*)]
\item Describe these three errors.
\item Write out a correct proof of the identity.\\
(b)[In this question take $g$ to be $9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }$ ]

\begin{figure}[h]
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\caption{Figure 3}
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Balls are projected,one after another,from a point,$A$ ,one metre above horizontal ground. Each ball travels in a vertical plane towards a 6 metre high vertical wall of negligible thickness,which is a horizontal distance of $10 \sqrt { 2 }$ metres from $A$ .

The balls are modelled as particles and it is assumed that there is no air resistance.\\
Each ball is projected with an initial speed of $28 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ and at a random angle $\theta$ to the horizontal,where $0 < \theta < 90 ^ { \circ }$

Given that a ball will pass over the wall precisely when $\alpha \leqslant \theta \leqslant \beta$
\item find, in degrees, the angle $\beta - \alpha$
\item Deduce that the probability that a particular ball will pass over the wall is $\frac { 2 } { 3 }$
\item Hence find the probability that exactly 2 of the first 10 balls projected pass over the wall.

You should give your answer in the form $\frac { P } { Q ^ { k } }$ where $P , Q$ and $k$ are integers and $P$ is not a multiple of $Q$.
\item Explain whether taking air resistance into account would increase or decrease the probability in (b)(iii).
\item find, in degrees, the angle $\beta - \alpha$
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2020 Q5 [22]}}