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Trig Proofs

A LevelAQAEdexcelOCR

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.

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Example 1

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

[3 marks]

$1 + cot ^2 A$

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

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

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

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

Example 2

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

[4 marks]

By $textcolor{limegreen}{cos} 2A equiv 2textcolor{limegreen}{cos} ^2 A – 1 equiv 1 – 2textcolor{blue}{sin} ^2 A$ ,

$sqrt{dfrac{1 – textcolor{limegreen}{cos} 2x}{1 + textcolor{limegreen}{cos} 2x}}$

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

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

$= textcolor{red}{tan} x$

Example 3

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

[2 marks]

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

$= textcolor{limegreen}{cos} (0 – x)$

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

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

$= textcolor{limegreen}{cos} x$

Example 4

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

[3 marks]

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

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

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

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

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

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

$= 2cosec 2 theta$

Trig Proofs Example Questions

Question 1: Show that, for small values of $x$ , $dfrac{3 – 3cos 2x}{4xtan x} approx dfrac{3}{2}$ .

[3 marks]

$dfrac{3 – 3(1 – 2x^2)}{4x^2} = dfrac{6x^2}{4x^2} = dfrac{3}{2}$

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Question 2: Prove that $dfrac{sin x}{cos x} = tan x$ .

[2 marks]

$sin x = dfrac{text{opp}}{text{hyp}}$

$cos x = dfrac{text{adj}}{text{hyp}}$

$tan x = dfrac{text{opp}}{text{adj}}$

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

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Question 3: Show that $cosec ^2 x – cos ^2 x equiv cot ^2 x + sin ^2 x$ .

[3 marks]

$begin{aligned}cot ^2 x + sin ^2 x&= dfrac{cos ^2 x}{sin ^2 x} + sin ^2 x[1.2em]&= dfrac{cos ^2 x}{sin ^2 x} + 1 – cos ^2 x[1.2em]&= dfrac{cos ^2 x}{sin ^2 x} + dfrac{sin ^2 x}{sin ^2 x} – cos ^2 x[1.2em]&= dfrac{cos ^2 x + sin ^2 x}{sin ^2 x} – cos ^2 x[1.2em]&=dfrac{1}{sin ^2 x} – cos ^2 x[1.2em]&= cosec ^2 x – cos ^2 xend{aligned}$