Trigonometric Proof (Cambridge (CIE) AS Maths: Pure 2): Exam Questions

By using the double angle formula for cosine, show that

$\text{cos}4\theta \equiv 8{\text{cos}}^4\theta -8{\text{cos}}^2\theta +1$

Show that

$\text{sin}\theta ({\text{cosec}}^2\theta -2)\equiv \frac{\text{cos}2\theta }{\text{sin}\theta }$

Show that

$\text{sin}3\theta +\text{sin}\theta \equiv 4\text{sin}\theta -4{\text{sin}}^3\theta$

Show that

$\frac{4\text{cot}x\text{cos}2x}{\text{sin}4x}\equiv {\text{cosec}}^{2}x$

where $x\ne \frac{k\pi }{4}$ and $k$ is an integer.

Show that

$\sqrt{2}\text{sin}\left(\theta -\frac{\pi }{4}\right)\equiv \text{sin}\theta -\text{cos}\theta$

Show that

$\text{sin}3\theta \equiv 3\text{sin}\theta -4{\text{sin}}^3\theta$

Show that

$\text{cos}4\theta +\text{cos}\frac{\pi }{3}\equiv 8{\text{sin}}^4\theta -8{\text{sin}}^2\theta +\frac{3}{2}$

Show that

$\text{tan}4\theta \equiv \frac{4\text{tan}\theta (1-{\text{tan}}^2\theta )}{1-6{\text{tan}}^2\theta +{\text{tan}}^4\theta }$

Show that

$\frac{\sqrt{2}\text{cos}\left(\theta +{\displaystyle \frac{\pi }{4}}\right)}{\text{sin}\left(\theta -{\displaystyle \frac{\pi }{2}}\right)}\equiv \text{tan}\theta -1$

Show that

${\text{cot}}^2\theta -{\text{tan}}^2\theta \equiv 4\text{cot}2\theta \text{cosec}2\theta$

Show that

$\frac{1-{\text{tan}}^{2}x}{\text{cos}2x}\equiv {\text{sec}}^{2}x$

where $x\ne \frac{(2k+1)\pi }{4}$ and $k$ is an integer.

Show that

$\text{cosec}x\equiv \frac{{\displaystyle \frac{1}{2}}{\text{sec}}^2{\displaystyle \frac{x}{2}}}{\text{tan}{\displaystyle \frac{x}{2}}}$

Show that

$\text{tan}\frac{x}{2}\equiv \frac{1}{\text{cosec}x+\text{cot}x}$

where $x\ne 2k\pi$ and $k$ is an integer.

Show that

$-16\text{cot}2\theta {\text{cosec}}^{3}2\theta \equiv {\text{sec}}^4\theta -{\text{cosec}}^4\theta$

Show that

$4{\text{cos}}^2\left(x-\frac{\pi }{6}\right)\equiv 3-2{\text{sin}}^{2}x+\sqrt{3}\text{sin}2x$

Show that

$\text{tan}\left(\frac{2x+\pi }{4}\right)\equiv \text{sec}x+\text{tan}x$

Show that

$\text{sin}3\theta \equiv 3\text{sin}\theta {\text{cos}}^2\theta -{\text{sin}}^3\theta$

Hence, or otherwise, show that

$\frac{\text{cos}3\theta -\text{cos}\theta }{\text{sin}3\theta \text{sin}\theta }\equiv \frac{4\text{cos}\theta }{1-4{\text{cos}}^2\theta }$

where $\theta \ne k\pi$ and $k$ is an integer.

Show that

$\frac{1}{{\left({\displaystyle \frac{\sqrt{3}}{2}}\text{cos}\theta -{\displaystyle \frac{1}{2}}\text{sin}\theta \right)}^2}+\frac{1}{{\left({\displaystyle \frac{\sqrt{3}}{2}}\text{sin}\theta +{\displaystyle \frac{1}{2}}\text{cos}\theta \right)}^2}\equiv 4{\text{cosec}}^2\left(2\theta +\frac{\pi }{3}\right)$