Chapter 15

Trigonometric identities

and equations

15.1 Trigonometric identities

A trigonometric identityis a relationship that is true
for all values of the unknown variable.

tanθ=

sinθ

cosθ

,cotθ=

cosθ

sinθ

,secθ=

1

cosθ

cosecθ=

1

sinθ

and cotθ=

1

tanθ

are examples of trigonometric identities from
Chapter 11.
Applying Pythagoras’ theorem to the right-angled
triangle shown in Fig. 15.1 gives:

a^2 +b^2 =c^2 (1)

b



a

c

Figure 15.1

Dividing each term of equation (1) byc^2 gives:

a^2

c^2

+

b^2

c^2

=

c^2

c^2

i.e.

(a

c

) 2

+

(

b

c

) 2

= 1

(cosθ)^2 +(sinθ)^2 = 1

Hence cos^2 θ+sin^2 θ=1(2)

Dividing each term of equation (1) bya^2 gives:

a^2

a^2

+

b^2

a^2

=

c^2

a^2

i.e. 1 +

(

b

a

) 2

=

(c

a

) 2

Hence 1 +tan^2 θ=sec^2 θ (3)

Dividing each term of equation (1) byb^2 gives:

a^2

b^2

+

b^2

b^2

=

c^2

b^2

i.e.

(a

b

) 2

+ 1 =

(c

b

) 2

Hence cot^2 θ+ 1 =cosec^2 θ (4)

Equations (2), (3) and (4) are three further examples

of trigonometric identities. For the proof of further

trigonometric identities, see Section 15.2.

15.2 Worked problems on

trigonometric identities

Problem 1. Prove the identity

sin^2 θcotθsecθ=sinθ.

With trigonometricidentitiesit is necessary to start with

the left-hand side (LHS) and attempt to make it equal to