Chapter 15
Trigonometric identities
and equations
15.1 Trigonometric identities
A trigonometric identity is a relationship that is true
for all values of the unknown variable.
tan θ =
sin θ
cos θ
, cot θ =
cos θ
sin θ
, sec θ =
1
cos θ
cosec θ =
1
sin θ
and cot θ =
1
tan θ
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) by c 2 gives:
a
2
c 2 +
b
2
c 2 =
c
2
c 2
i.e.
a
c
2 +
b
c
2
= 1
(cos θ) 2 + (sin θ) 2 = 1
Hence
cos
2 θ + sin
2 θ = 1
( 2 )
Dividing each term of equation (1) by a 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) by b 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 θ cot θ sec θ = sin θ.
With trigonometric identities it is necessary to start with
the left-hand side (LHS) and attempt to make it equal to