Proving identities

Trigonometry, grade 11 (5 units) Lesson 7 of 18Learning

Proving identities

What an identity is

An identity is an equality that holds for every $\displaystyle x$ where both sides are defined. For example, $\displaystyle {\tan x\cdot\cos x}={\sin x}$ holds for every $\displaystyle x$ with $\displaystyle \cos x\neq0$ .

A proof: a chain of equalities that starts at one side and reaches the other. Each equality in the chain rests on a known identity or on algebra.
Which side to start from: usually the more complicated one – simplifying is easier than complicating.
Disproving: one $\displaystyle x$ where the sides differ is enough. For example, $\displaystyle {\sin2x}={2\sin x}$ is no identity: at $\displaystyle {x}={\frac{\pi}{2}}$ the left side is $\displaystyle 0$ and the right side $\displaystyle 2$ .

A common mistake in a proof

Careful: do not write the identity to be proved and then act on both sides until you reach something true. That proves only the converse: the true statement follows from the identity – not the other way round. An example: from $\displaystyle {1}={-1}$ , which is false, squaring gives $\displaystyle {1}={1}$ , which is true.

Ways to start

Write everything with $\displaystyle \sin$ and $\displaystyle \cos$ ( $\displaystyle {\tan x}={\frac{\sin x}{\cos x}}$ ).
Bring fractions to a common denominator.
Replace $\displaystyle 1$ by $\displaystyle \sin^2x+\cos^2x$ , or the other way round.
Open a double angle when the other side has $\displaystyle x$ alone, and the other way round.
Multiply the numerator and the denominator by the conjugate: $\displaystyle {(1-\cos x)(1+\cos x)}={\sin^2x}$ .
Factorise: a difference of two squares, a common factor.

An example – a common denominator

A proof of $\displaystyle {\frac{1}{\cos x}-\cos x}={\sin x\tan x}$ , from the left side:

\frac{1}{\cos x}-\cos x=\frac{1-\cos^2x}{\cos x}=\frac{\sin^2x}{\cos x}=\sin x\cdot\frac{\sin x}{\cos x}=\sin x\tan x

An example – the conjugate

A proof of $\displaystyle {\frac{\sin x}{1-\cos x}}={\frac{1+\cos x}{\sin x}}$ : multiply the numerator and the denominator of the left side by $\displaystyle 1+\cos x$ .

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

Exercise 1

Prove: $\displaystyle {\tan x+\frac{1}{\tan x}}={\frac{2}{\sin2x}}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Prove: $\displaystyle {(\sin x+\cos x)^2}={1+\sin2x}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Prove: $\displaystyle {\frac{\cos2x}{\cos x+\sin x}}={\cos x-\sin x}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

Prove: $\displaystyle {\frac{\sin2x}{\sin x}-\frac{\cos2x}{\cos x}}={\frac{1}{\cos x}}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 5

Prove: $\displaystyle {\sin\left(\frac{\pi}{4}+x\right)\sin\left(\frac{\pi}{4}-x\right)}={\frac{1}{2}\cos2x}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 6

Prove: $\displaystyle {\frac{\sin3x}{\sin x}-\frac{\cos3x}{\cos x}}={2}$

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 7

Is $\displaystyle {\cos2x}={2\cos x-1}$ an identity? If so, prove it; if not, give an $\displaystyle x$ where the sides differ.

Formulas: trigonometry, grade 11 (5 units)

Basic identities

The Pythagorean identity
$\displaystyle \sin^{2}\alpha+\cos^{2}\alpha$ $\displaystyle =$ $\displaystyle 1$
The tangent
$\displaystyle \tan\alpha$ $\displaystyle =$ $\displaystyle \frac{\sin\alpha}{\cos\alpha}$ $\displaystyle \cos\alpha\neq0$
$\displaystyle 1+\tan^{2}\alpha$ $\displaystyle =$ $\displaystyle \frac{1}{\cos^{2}\alpha}$
$\displaystyle \sin(-\alpha)$ $\displaystyle =$ $\displaystyle -\sin\alpha$
$\displaystyle \cos(-\alpha)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \sin(\pi-\alpha)$ $\displaystyle =$ $\displaystyle \sin\alpha$
$\displaystyle \cos(\pi-\alpha)$ $\displaystyle =$ $\displaystyle -\cos\alpha$
$\displaystyle \sin\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \cos\alpha$
$\displaystyle \cos\left(\frac{\pi}{2}-\alpha\right)$ $\displaystyle =$ $\displaystyle \sin\alpha$

The sum and the difference of angles

$\displaystyle \cos(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \cos\alpha\cdot\cos\beta\mp\sin\alpha\cdot\sin\beta$
$\displaystyle \sin(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \sin\alpha\cdot\cos\beta\pm\cos\alpha\cdot\sin\beta$
$\displaystyle \tan(\alpha\pm\beta)$ $\displaystyle =$ $\displaystyle \frac{\tan\alpha\pm\tan\beta}{1\mp\tan\alpha\cdot\tan\beta}$

The double angle

$\displaystyle \sin2\alpha$ $\displaystyle =$ $\displaystyle 2\sin\alpha\cdot\cos\alpha$
$\displaystyle \cos2\alpha=\cos^{2}\alpha-\sin^{2}\alpha=2\cos^{2}\alpha-1=1-2\sin^{2}\alpha$
$\displaystyle \tan2\alpha$ $\displaystyle =$ $\displaystyle \frac{2\tan\alpha}{1-\tan^{2}\alpha}$
$\displaystyle \cos^{2}\alpha$ $\displaystyle =$ $\displaystyle \frac{1+\cos2\alpha}{2}$
$\displaystyle \sin^{2}\alpha$ $\displaystyle =$ $\displaystyle \frac{1-\cos2\alpha}{2}$

Sums and differences of sines and of cosines

$\displaystyle \sin\alpha+\sin\beta$ $\displaystyle =$ $\displaystyle 2\sin\frac{\alpha+\beta}{2}\cdot\cos\frac{\alpha-\beta}{2}$
$\displaystyle \sin\alpha-\sin\beta$ $\displaystyle =$ $\displaystyle 2\sin\frac{\alpha-\beta}{2}\cdot\cos\frac{\alpha+\beta}{2}$
$\displaystyle \cos\alpha+\cos\beta$ $\displaystyle =$ $\displaystyle 2\cos\frac{\alpha+\beta}{2}\cdot\cos\frac{\alpha-\beta}{2}$
$\displaystyle \cos\alpha-\cos\beta$ $\displaystyle =$ $\displaystyle -2\sin\frac{\alpha+\beta}{2}\cdot\sin\frac{\alpha-\beta}{2}$

Equations – the general solution ($k$ a whole number)

$\displaystyle \sin x=\sin\alpha\iff x=\alpha+2\pi k\ \text{or}\ x=\pi-\alpha+2\pi k$
$\displaystyle \cos x=\cos\alpha\iff x=\pm\alpha+2\pi k$
$\displaystyle \tan x=\tan\alpha\iff x=\alpha+\pi k$

A triangle

The law of sines ( $\displaystyle R$ – the radius of the circumscribed circle)
$\displaystyle \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R$
The law of cosines ( $\displaystyle \gamma$ – the angle between $\displaystyle a$ and $\displaystyle b$ )
$\displaystyle c^{2}$ $\displaystyle =$ $\displaystyle a^{2}+b^{2}-2a\cdot b\cdot\cos\gamma$
The area of a triangle ( $\displaystyle \alpha$ – the angle between $\displaystyle b$ and $\displaystyle c$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}b\cdot c\cdot\sin\alpha$
The area of a triangle ( $\displaystyle \beta$ , $\displaystyle \gamma$ – the angles next to $\displaystyle a$ )
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{a^{2}\cdot\sin\beta\cdot\sin\gamma}{2\cdot\sin(\beta+\gamma)}$

A circle ($\alpha$ in radians, $R$ – the radius)

The area of a disc
$\displaystyle S$ $\displaystyle =$ $\displaystyle \pi R^{2}$
The circumference of a circle
$\displaystyle P$ $\displaystyle =$ $\displaystyle 2\pi R$
The length of an arc
$\displaystyle l$ $\displaystyle =$ $\displaystyle \alpha\cdot R$
The area of a sector
$\displaystyle S$ $\displaystyle =$ $\displaystyle \frac{1}{2}\alpha\cdot R^{2}$

Solids ($S$ – the base area, $h$ – the height, $l$ – the slant height)

The volume of a prism and a cylinder
$\displaystyle V$ $\displaystyle =$ $\displaystyle S\cdot h$
The volume of a pyramid and a cone
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{S\cdot h}{3}$
The lateral surface of a cone
$\displaystyle M$ $\displaystyle =$ $\displaystyle \pi R\cdot l$
The volume of a sphere
$\displaystyle V$ $\displaystyle =$ $\displaystyle \frac{4}{3}\pi R^{3}$
The surface area of a sphere
$\displaystyle M$ $\displaystyle =$ $\displaystyle 4\pi R^{2}$