Trigonometric Proof (AQA A Level Maths: Pure): Exam Questions

Prove that

$2\mathrm{cosec}\left(2x\right)\cot x\equiv {\mathrm{cosec}}^{2}x$

where $x\ne \frac{n\pi }{2},n\in \mathrm{\mathbb{Z}}$.

Prove that

${\sin }^2\theta \left({\sec }^2\theta +{\mathrm{cosec}}^2\theta \right)\equiv {\sec }^2\theta$

Prove that

$\frac{4{\sin }^4\theta }{{\sin }^2\left(2\theta \right)}\equiv {\tan }^2\theta$

where $x\ne k\pi ,k\in \mathrm{\mathbb{Z}}$.

Show that

$\sin \theta \left(-2+{\mathrm{cosec}}^2\theta \right)\equiv \frac{\cos \left(2\theta \right)}{\sin \theta }$

Show that

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

Prove that

$\frac{4\cot x\cos \left(2x\right)}{\sin \left(4x\right)}\equiv {\mathrm{cosec}}^{2}x$

where $x\ne \frac{k\pi }{4},k\in \mathrm{\mathbb{Z}}$.

Prove that

$\frac{1-{\tan }^{2}x}{\cos \left(2x\right)}\equiv {\sec }^{2}x$

where $x\ne \left(\frac{2k+1}{4}\right)\pi ,k\in \mathrm{\mathbb{Z}}$.

Given that

• $\theta$ is small and in radians

• terms involving ${\theta }^3$ (or higher powers of $\theta$) can be ignored

show that

$4\cos \left(4\theta \right)-2{\cos }^2\left(2\theta \right)\approx a+b{\theta }^2$

where $a$ and $b$ are constants to be found.

Show that the result in part (a) gives a percentage error of 0.583%, to 3 significant figures, when used to approximate

$4\cos \left(\frac{\pi }{6}\right)-2{\cos }^2\left(\frac{\pi }{12}\right)$

(i) Prove the identity

$\sin \left(3\theta \right)\equiv 3\sin \theta -4{\sin }^3\theta$

(ii) Use a counter example to disprove the statement

$\cos \left(3\theta \right)\equiv 3\cos \theta -4{\cos }^3\theta$

Prove that

${\cot }^2\theta -{\tan }^2\theta \equiv 4\cot \left(2\theta \right)\mathrm{cosec}\left(2\theta \right)$

Prove the identity

$\frac{1}{\sin 2\theta }-\cot 2\theta \equiv \tan \theta ,\theta \ne \frac{n\pi }{2}$

Prove that

$\mathrm{cosec}x\equiv \frac{{\displaystyle \frac{1}{2}}{\sec }^2\left({\displaystyle \frac{x}{2}}\right)}{\tan {\displaystyle \left(\frac{x}{2}\right)}}$

Use the identity

$\tan x\equiv \frac{\sin x}{\cos x}$

to show that

$\frac{\mathrm{d}}{\mathrm{d}x}\left[\tan x\right]\equiv {\sec }^{2}x$

Show that

$\frac{\mathrm{d}}{\mathrm{d}x}\left[\mathrm{cosec}x\right]\equiv -\cot x\mathrm{cosec}x$

Use the identity

$\sin A+\sin B\equiv 2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)$

to show that

$\sin \left(3\theta \right)+\sin \theta \equiv p\sin \theta +q{\sin }^3\theta$

where $p$ and $q$ are constants to be found.

Given that $\theta$ is small, and that terms involving ${\theta }^3$ or higher powers of $\theta$ can be ignored, show that

$\frac{1}{{\mathrm{cosec}}^2\left({\displaystyle \frac{\theta }{2}}\right)}+\frac{1}{{\sec }^2\left({\displaystyle \frac{\theta }{4}}\right)}\approx 1+\frac{3}{16}{\theta }^2$

Find the percentage error when the result in part (a) is used to approximate

$\frac{1}{{\mathrm{cosec}}^2\left({\displaystyle \frac{7}{20}}\right)}+\frac{1}{{\sec }^2\left({\displaystyle \frac{7}{40}}\right)}$

giving your answer correct to 3 significant figures.

Show that

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

Show that

$\frac{10}{4\cos \theta +3\sin \theta }\equiv 2\sec \left(\theta -\mathrm{\alpha }\right)$

where

$\alpha =\arctan \left(\frac{3}{4}\right)$

Show that

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

Prove that

$-16\cot \left(2\theta \right){\mathrm{cosec}}^3\left(2\theta \right)\equiv {\sec }^4\theta -{\mathrm{cosec}}^4\theta$

(i) Show that

$\cos \left(4\theta \right)\equiv p{\cos }^4\theta +q{\cos }^2\theta +r$

where $p$, $q$ and $r$ are integers to be found.

(ii) Use a counter example to disprove the statement

$\sin \left(4\theta \right)\equiv p{\sin }^4\theta +q{\sin }^2\theta +r$

where $p$, $q$ and $r$ are their values from part (i).

Prove that

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

Show that

$\tan \left(\frac{x}{2}\right)\equiv \frac{1}{\mathrm{cosec}x+\cot x}$

where $x\ne 2k\pi ,k\in \mathrm{\mathbb{Z}}$.

Show that

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

Show that

$\cos \left(4\theta \right)+\cos \left(\frac{\pi }{3}\right)\equiv 8{\sin }^4\theta -8{\sin }^2\theta +\frac{3}{2}$

Consider the three triangles, all of height 1, as shown below.

By applying the area formula $\frac{1}{2}ab\sin C$  to each one, prove the identity

$\sin \left(A+B\right)\equiv \sin A\cos B+\sin B\cos A$

Suggest one limitation of this proof.

Show that

$\sin \left(3\theta \right)\equiv 3\sin \theta {\cos }^2\theta -{\sin }^3\theta$

Hence, or otherwise, show that

$\frac{\cos \left(3\theta \right)-\cos \theta }{\sin \left(3\theta \right)\sin \theta }\equiv \frac{4\cos \theta }{1-4{\cos }^2\theta }$

where $\theta \ne k\pi ,k\in \mathrm{\mathbb{Z}}.$

Prove that

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