Cambridge International AS and A Level Mathematics Pure Mathematics 1

Trigonometry

P1^

7

an equation: it is true when θ = 45° or 225°, but not when it takes any other value

in the range 0°  θ  360°.

By contrast, an identity is true for all values of the variable, for example

tan sin

cos

,tan sin

cos

30 30 ,tan(–

30

72 72

72

°= ° 33

°

°= °

°

99 399

399

°= °

°

) sin(–)

cos(–)

,

and so on for all values of the angle.

In this book, as in mathematics generally, we often use an equals sign where

it would be more correct to use an identity sign. The identity sign is kept for

situations where we really want to emphasise that the relationship is an identity

and not an equation.

Another useful identity can be found by applying Pythagoras’ theorem to any

point P(x, y) on the unit circle

y^2 + x^2 ≡     OP^2

(sin θ)^2 + (cos θ)^2 ≡ 1.

This is written as

sin^2 θ + cos^2 θ ≡ 1.

You can use the identities tan sin

cos

θ θ

θ

≡ and sin^2 θ + cos^2 θ ≡ 1 to prove other

identities are true.

There are two methods you can use to prove an identity; you can use either

method or a mixture of both.

Method 1

When both sides of the identity look equally complicated you can work with

both the left-hand side (LHS) and the right-hand side (RHS) and show that

LHS – RHS = 0.

ExAmPlE 7.3 Prove the identity cos^2 θ – sin^2 θ ≡ 2 cos^2 θ – 1.

SOlUTION

Both sides look equally complicated, so show LHS – RHS = 0.

So you need to show cos^2 θ – sin^2 θ – 2 cos^2 θ + 1 ≡ 0.

Simplifying:

cos^2 θ – sin^2 θ – 2 cos^2 θ + 1 ≡ – cos^2 θ – sin^2 θ + 1

≡ –(cos^2 θ + sin^2 θ) + 1

≡ –1 + 1 Using sin (^2) θ + cos (^2) θ = 1.
≡ 0 as required