Trig Proofs

Trig Proofs

A LevelAQAEdexcelOCREdexcel 2022OCR 2022

Trig Proofs

In this section, we’ll be taking a look at some standard proof methods, involving all of the techniques we’ve learned so far.

A Level AQA Edexcel OCR

Example 1

Prove that 1 + \cot ^2 A \equiv \cosec ^2 A.

[3 marks]

1 + \cot ^2 A

= \dfrac{\textcolor{blue}{\sin} ^2 \textcolor{purple}{A}}{\textcolor{blue}{\sin} ^2 \textcolor{purple}{A}} + \dfrac{\textcolor{limegreen}{\cos} ^2 \textcolor{purple}{A}}{\textcolor{blue}{\sin} ^2 \textcolor{purple}{A}}

= \dfrac{\textcolor{blue}{\sin} ^2 \textcolor{purple}{A} + \textcolor{limegreen}{\cos} ^2 \textcolor{purple}{A}}{\textcolor{blue}{\sin} ^2 \textcolor{purple}{A}}

= \dfrac{1}{\textcolor{blue}{\sin} ^2 \textcolor{purple}{A}}

= \cosec ^2 \textcolor{purple}{A}

A LevelAQAEdexcelOCR

Example 2

Show that \sqrt{\dfrac{1 - \textcolor{limegreen}{\cos 2x}}{1 + \textcolor{limegreen}{\cos} 2x}} = \textcolor{red}{\tan} x.

[4 marks]

By \textcolor{limegreen}{\cos} 2A \equiv 2\textcolor{limegreen}{\cos} ^2 A - 1 \equiv 1 - 2\textcolor{blue}{\sin} ^2 A,

\sqrt{\dfrac{1 - \textcolor{limegreen}{\cos} 2x}{1 + \textcolor{limegreen}{\cos} 2x}}

= \sqrt{\dfrac{2\textcolor{blue}{\sin} ^2 x}{2\textcolor{limegreen}{\cos} ^2 x}}

= \sqrt{\textcolor{red}{\tan} ^2 x}

= \textcolor{red}{\tan} x

A LevelAQAEdexcelOCR

Example 3

Prove that \textcolor{limegreen}{\cos} x = \textcolor{limegreen}{\cos} (-x).

[2 marks]

\textcolor{limegreen}{\cos} (-x)

= \textcolor{limegreen}{\cos} (0 - x)

= \textcolor{limegreen}{\cos} 0 \textcolor{limegreen}{\cos} x + \textcolor{blue}{\sin} 0 \textcolor{blue}{\sin} x

= (1 \times \textcolor{limegreen}{\cos} x) + (0 \times \textcolor{blue}{\sin} x)

= \textcolor{limegreen}{\cos} x

A LevelAQAEdexcelOCR

Example 4

Show that \textcolor{red}{\tan} \theta + \cot \theta = 2\cosec 2 \theta.

[3 marks]

\textcolor{red}{\tan} \theta + \cot \theta

= \dfrac{\textcolor{blue}{\sin} \theta}{\textcolor{limegreen}{\cos} \theta} + \dfrac{\textcolor{limegreen}{\cos} \theta}{\textcolor{blue}{\sin} \theta}

= \dfrac{\textcolor{blue}{\sin} ^2 \theta}{\textcolor{blue}{\sin} \theta \textcolor{limegreen}{\cos} \theta} + \dfrac{\textcolor{limegreen}{\cos} ^2 \theta}{\textcolor{blue}{\sin} \theta \textcolor{limegreen}{\cos} \theta}

= \dfrac{\textcolor{blue}{\sin} ^2 \theta + \textcolor{limegreen}{\cos} ^2 \theta}{\textcolor{blue}{\sin} \theta \textcolor{limegreen}{\cos} \theta}

= \dfrac{1}{\textcolor{blue}{\sin} \theta \textcolor{limegreen}{\cos} \theta}

= \dfrac{2}{\textcolor{blue}{\sin} 2\theta}

= 2\cosec 2 \theta

A LevelAQAEdexcelOCR

Example Questions

\dfrac{3 - 3(1 - 2x^2)}{4x^2} = \dfrac{6x^2}{4x^2} = \dfrac{3}{2}

  • \sin x = \dfrac{\text{opp}}{\text{hyp}}

 

  • \cos x = \dfrac{\text{adj}}{\text{hyp}}

 

  • \tan x = \dfrac{\text{opp}}{\text{adj}}

 

\dfrac{\sin x}{\cos x} = \dfrac{\left( \dfrac{\text{opp}}{\text{hyp}}\right) }{\left( \dfrac{\text{adj}}{\text{hyp}}\right) } = \dfrac{\text{opp}}{\text{adj}} = \tan x

\begin{aligned}\cot ^2 x + \sin ^2 x&= \dfrac{\cos ^2 x}{\sin ^2 x} + \sin ^2 x\\[1.2em]&= \dfrac{\cos ^2 x}{\sin ^2 x} + 1 - \cos ^2 x\\[1.2em]&= \dfrac{\cos ^2 x}{\sin ^2 x} + \dfrac{\sin ^2 x}{\sin ^2 x} - \cos ^2 x\\[1.2em]&= \dfrac{\cos ^2 x + \sin ^2 x}{\sin ^2 x} - \cos ^2 x\\[1.2em]&=\dfrac{1}{\sin ^2 x} - \cos ^2 x\\[1.2em]&= \cosec ^2 x - \cos ^2 x\end{aligned}

Additional Resources

MME

Exam Tips Cheat Sheet

A Level
MME

Formula Booklet

A Level

Worksheet and Example Questions

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Trigonometry

A Level

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