Compound angle identities, other HL trig identities: notes and practice questions

(a) Show that $\cos(4\theta) = 8\cos^4(\theta) - 8\cos^2(\theta) + 1$.

(b) Hence, solve the equation $16\cos^4(\theta) - 16\cos^2(\theta) + 1 = 0$ for $0 \le \theta \le \pi$.

Show that $arctan(3) - arctan\left(\frac{1}{2}\right) = \frac{\pi}{4}$.

Hence, or otherwise, find the exact value of $arctan(3) + arctan(2)$.

The following diagram shows the graph of $y = \arctan\left(\frac{x-3}{2}\right) -\frac{\pi}{4}$ for $x \in \mathbb{R}$, with asymptotes at $y = -\frac{3\pi}{4}$ and $y = \frac{\pi}{4}$.

Describe a sequence of transformations that transforms the graph of $y = \arctan x$ to the graph of $y = \arctan\left(\frac{x-3}{2}\right) -\frac{\pi}{4}$ for $x \in \mathbb{R}$.

Show that $\arctan p + \arctan q = \arctan\left(\frac{p+q}{1-pq}\right)$ where $p, q > 0$ and $pq < 1$.

Using mathematical induction and the result from part (b), prove that

$\sum_{r=1}^{n} \arctan\left(\frac{1}{r^2+r+1}\right) = \arctan\left(\frac{n}{n+2}\right)$ for $n \in \mathbb{Z}^+$.

(a) Solve the equation $\cos(x - 45^\circ) = \sqrt{2} \sin(x)$ for $0^\circ \le x \le 360^\circ$.

(b) Given that $\tan(A + B) = \frac{1}{3}$ and $\tan(B) = 2$, find the exact value of $\tan(A)$.

Prove that $\cot\left(\theta + \frac{\pi}{4}\right) = \frac{1 - \sin 2\theta}{\cos 2\theta}$, where $\theta \ne \frac{\pi}{4} + \frac{n\pi}{2}, n \in \mathbb{Z}$.

Hence, solve the equation $\frac{1 - \sin x}{\cos x} = 1$ for $0 \le x \le 2\pi$.

Given that $\pi \le \theta \le \frac{3\pi}{2}$ and $\sin \theta = -\frac{8}{17}$, find the value of $\cos \theta$.

Find the value of $\sin 2\theta$.

Find the value of $\cos(\theta + \frac{\pi}{3})$.