Use the principle of mathematical induction to prove that

$\sin x+\sin 3x+\dots +\sin (2n-1)x=\frac{1-\cos 2nx}{2\sin x},n\in {\mathbb{Z}}^{+},x\ne k\pi$ where $k\in \mathbb{Z}$.

Find $\cos \theta$.

Hence or otherwise solve $\text{lo}{\text{g}}_3\left(\text{2}\text{sin}x\right)=\text{lo}{\text{g}}_9\left(\text{cos}2x+2\right)$ for .

Show that $\frac{1-\cos 2x}{2\sin x}\equiv \sin x,x\ne k\pi$ where $k\in \mathbb{Z}$.

Given that $\tan \theta >0$, find $\tan \theta$.

Given that $\text{cos}\hat{A}=\frac{5}{6}$ find the value of $\text{sin}\hat{A}$.

Find the value of $\text{tan}\theta$.

Let $y=\frac{1}{\cos x}$, for . The graph of $y$between $x=\theta$ and $x=\frac{\pi }{4}$ is rotated 360° about the $x$-axis. Find the volume of the solid formed.

Show that $\text{lo}{\text{g}}_9\left(\text{cos}2x+2\right)=\text{lo}{\text{g}}_3\sqrt{\text{cos}2x+2}$.

Solve the equation $(f\circ g)\left(x\right)=2 \cos 2x$ where $0\le x\le \pi$.

By using a suitable substitution for $a$, show that $1-3 \cos 2x+3 {\cos }^2 2x- {\cos }^3 2x=8 {\sin }^6 x$.

Given that $\text{A}\hat{\text{B}}\text{C}$ is acute, find $\sin \theta$.

Find $\cos \left(2\times \text{C}\hat{\text{A}}\text{B}\right)$.

Find the area of triangle ABC.

Show that $\sin 2x+\cos 2x-1=2 \sin x\left(\cos x-\sin x\right)$.

Solve ${\log }_2(2\sin x)+{\log }_2(\cos x)=-1$, for .

Given that $\text{sin}x=\frac{1}{3}$, where , find the value of $\text{cos}4x$.

Hence or otherwise solve the equation $\sin x+\sin 3x=\cos x$ in the interval .

Show that $f\text{'}\text{'}\left(x\right)=-\frac{1}{4\sqrt{{\left(1+x\right)}^3}}$.

Find $\cos 2\theta$.

Show that $\cos \theta =\frac{3}{4}$.

Hence or otherwise, solve $\sin 2x+\cos 2x-1+\cos x-\sin x=0$ for $0<x<2\mathrm{\pi }$.

Show that the equation $2 {\cos }^2 x+5 \sin x=4$ may be written in the form $2 {\sin }^2 x-5 \sin x+2=0$.