Higher Engineering Mathematics

Geometry and trigonometry

16

Trigonometric identities and equations

16.1 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 12.

Applying Pythagoras’ theorem to the right-angled

triangle shown in Fig. 16.1 gives:

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

Figure 16.1

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 16.2.

16.2 Worked problems on

trigonometric identities

Problem 1. Prove the identity

sin^2 θcotθsecθ=sinθ.

With trigonometric identities it is necessary to start

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

it equal to the right-hand side (RHS) or vice-versa.

It is often useful to change all of the trigonometric

ratios into sines and cosines where possible. Thus,

LHS=sin^2 θcotθsecθ

=sin^2 θ

(

cosθ

sinθ

)(

1

cosθ

)

=sinθ(by cancelling)=RHS

Problem 2. Prove that

tanx+secx

secx

(

1 +

tanx

secx

)= 1.