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