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．
（i）Describe these three errors．
（ii）Write out a correct proof of the identity．
（b）［In this question take \(g\) to be \(9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) ］
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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 \(\alpha \leqslant \theta \leqslant \beta\)
- find, in degrees, the angle \(\beta - \alpha\)
- Deduce that the probability that a particular ball will pass over the wall is \(\frac { 2 } { 3 }\)
- 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 \(Q\).
- Explain whether taking air resistance into account would increase or decrease the probability in (b)(iii).
- find, in degrees, the angle \(\beta - \alpha\)