Answer: (a) \(k=6\) or \(k=14\); (b)(ii) \(\theta=213.3^\circ\) or \(326.7^\circ\).

Answer: (a) \(k=6\) or \(k=14\); (b)(ii) \(\theta=213.3^\circ\) or \(326.7^\circ\).

(a) Since \(-1\le\sin\left(x+\frac\pi4\right)\le1\), the range of the curve is

\(6\le y\le14.\)

So the horizontal tangents occur at \(k=6\) and \(k=14\).

(b)(i) Use the common denominator \((1-\cos\theta)(1+\cos\theta)=\sin^{2}\theta\):

\(\frac{(1+\tan\theta)(1+\cos\theta)+(1-\tan\theta)(1-\cos\theta)}{\sin^{2}\theta}.\)

The numerator expands to

\(2+2\tan\theta\cos\theta.\)

Since \(\tan\theta\cos\theta=\sin\theta\), this becomes

\(\frac{2(1+\sin\theta)}{\sin^{2}\theta}.\)

(b)(ii) Hence

\(\frac{2(1+\sin\theta)}{\sin^{2}\theta}=3.\)

So

\(3\sin^{2}\theta-2\sin\theta-2=0.\)

Solving gives

\(\sin\theta=\frac{1\pm\sqrt7}{3}.\)

The positive value is greater than \(1\), so reject it. Hence

\(\sin\theta=\frac{1-\sqrt7}{3}\approx-0.5486.\)

This gives

\(\theta=213.3^\circ\quad\text{or}\quad326.7^\circ.\)