Proving trigonometric identities

​​In a nutshell

You can use all the trigonometric identities that you have learned so far to prove more trigonometric identities.

Proving trigonometric identities

Proofs may involve using the reciprocal trigonometric functions, the addition formulae and double angle formulae.

Example 1

Prove that

​$$\tan x + \cot x \equiv 2 \cosec 2x$$​​

Start from the left hand side and change all trig functions into $$\sin$$ and $$\cos$$:

$$\begin{aligned}\tan x + \cot x &= \dfrac{\sin x}{\cos x} +\dfrac{\cos x}{\sin x} \\\\&= \dfrac{\sin^2x + \cos^2x}{\sin x \cos x} \\\ \end{aligned}$$​​

Use $$\sin^2x + \cos^2x=1$$​

​$$\begin{aligned} \dfrac{\sin^2x + \cos^2x}{\sin x \cos x} & = \dfrac{1}{\sin x \cos x} \\\\& = \dfrac{2}{2 \sin x \cos x} \\ \end{aligned}$$​​

Use the double angle formula $$\sin 2x = 2 \sin x \cos x$$:

​$$\begin{aligned} \dfrac{2}{2 \sin x \cos x} & = \dfrac{2}{ \sin 2x } \\\\&= 2 \cosec 2x\end{aligned}$$​​

Therefore, $$\underline{\tan x + \cot x \equiv 2 \cosec 2x}$$.

Example 2

Prove that

​$$\cos 3x \equiv 4 \cos^3 x - 3 \cos x$$​​

Start from the left hand side and write $$\cos 3x$$ as $$\cos (2x+x)$$ and use the addition formula:

$$\begin{aligned}\cos 3x &= \cos (2x+x) \\&= \cos 2x \cos x - \sin 2x \sin x \\\end{aligned}$$​​

Use the double angle formulae to give:

​$$\begin{aligned} \cos 2x \cos x - \sin 2x \sin x &= (2\cos^2x-1) \cos x - 2\sin x \cos x \sin x \\&= 2 \cos^3 x - \cos x - 2 \cos x \sin^2 x \\\end{aligned}$$​​

Use $$\cos^2x + \sin^2 x=1$$ to convert to $$\cos$$:

​$$\begin{aligned} 2 \cos^3 x - \cos x - 2 \cos x \sin^2 x &= 2 \cos^3 x - \cos x - 2 \cos x (1- \cos^2 x) \\& = 2 \cos^3x - \cos x - 2 \cos x + 2 \cos^3 x \\&= 4 \cos^3 x - 3 \cos x \end{aligned}$$​​

Therefore, $$\underline{\cos 3x \equiv 4 \cos^3 x - 3 \cos x}$$.