Trigonometric Identities (Edexcel IGCSE Further Pure Maths): Revision Note

• When asked to show that one thing is equal or identical to another, look at what parts are 'missing'

  • This can help you spot which identity to use

Show that the equation $2{\sin }^2\theta -\cos \theta =0$ can be written in the form  $a{\cos }^2\theta +b\cos \theta +c=0$, where $a$, $b$ and $c$ are integers to be found with $a>0$.

Note that in the 'target' form there is no $\sin \theta$ or ${\sin }^2\theta$

That means we want to use a substitution to get rid of the ${\sin }^2\theta$ in the original form

We can do this using the identity ${\sin }^2\theta +{\cos }^2\theta =1$, rearranged as  ${\sin }^2\theta =1-{\cos }^2\theta$

$\begin{array}{rcl}2{\sin }^2\theta -\cos \theta & =& 2\left(1-{\cos }^2\theta \right)-\cos \theta \\ & =& 2-2{\cos }^2\theta -\cos \theta \\ & =& -2{\cos }^2\theta -\cos \theta +2\end{array}$

Substitute that back into the original equation

$\begin{array}{rcl}-2{\cos }^2\theta -\cos \theta +2& =& 0\end{array}$

Multiply both sides of the equation by $-1$ to make the ${\cos }^2\theta$ coefficient positive

This gets the equation into the required form with  $a=2$, $b=1$ and $c=-2$

$\mathbf{2}{\mathbf{cos}}^{\mathbf{2}}\mathbf{\theta }\mathbf{+}\mathbf{cos}\mathbf{\theta }\mathbf{-}\mathbf{2}\mathbf{=}\mathbf{0}$