Trigonometric Proof

You can use trigonometric identities you already know to prove new identities
Make sure you know the simple trigonometric identities and further trigonometric identities
- $\sin^{2} θ + \cos^{2} θ = 1$
- $\frac{\sin θ}{\cos θ} = \tan θ$
- $1 + \tan^{2} θ = \sec^{2} θ$
- $1 + \cot^{2} θ = {cosec}^{2} θ$
To prove an identity start on one side and proceed step by step until you get to the other side
- e.g. to prove that ${(\sin θ + \cos θ)}^{2} = 1 + 2 \sin θ \cos θ$
- Start with the left hand side, and expand
  - $\sin^{2} θ + \cos^{2} θ + 2 \sin θ \cos θ$
- Simplify using the identity $\sin^{2} θ + \cos^{2} θ = 1$
  - $1 + 2 \sin θ \cos θ$
- This is now equal to the expression on the right, so the statement has been proven

$Example including a fraction within a fraction, and cot$

Always keep an eye on the 'target' expression – this can help suggest what identities to use

Don't forget that you can start a proof from either side – sometimes it might be easier to start with a particular side
- If you get stuck - try starting from the other side instead!

Prove that:

$\sec^{2} x (\cot^{2} x - \cos^{2} x) = \cot^{2} x$

Use the definitions of $\sec x = \frac{1}{\cos x}$ and $\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$ , to rewrite the left hand side

$\frac{1}{\cos^{2} x} (\frac{\cos^{2} x}{\sin^{2} x} - \cos^{2} x)$

Expand the brackets, using the fact that multiplying by $\frac{1}{\cos^{2} x}$ is the same as dividing by $\cos^{2} x$

$\frac{1}{\sin^{2} x} - 1$

Use the definition of $cosec x = \frac{1}{\sin x}$ to rewrite the expression

${cosec}^{2} x - 1$

Use the identity $1 + \cot^{2} x = {cosec}^{2} x$ to rewrite the expression

$1 + \cot^{2} x - 1$

Simplify to achieve the right hand side

$\cot^{2} x$

Trigonometric Proof (CIE IGCSE Additional Maths)