Trigonometric Identities

Trigonometric Identities

An identity is an equation that is satis...ed by all the values of the variable(s) in the equation. We have already introduced the following:

sin x (a) tan x =

cos x

1 (b) sec x =

cos x

1 (c) csc x =

sin x

1 (d) cot x =

tan x

The trigonometric identity sin2 x + cos2 x = 1

The trig identity sin2 x + cos2 x = 1 follows from the Pythagorean theorem. In the ...gure below, an angle x is drawn. The side opposite the angle has length a, the adjacent side has length b, and the hypotenuse has length h.

h a

x b

a

b

Therefore, sin x = and cos x = . It follows that

h

h

(sin x)2 + (cos x)2 =

a2 +

h

b 2 a2 b2 a2 + b2 h = h2 + h2 = h2

The Pythagorean theorem asserts that a2 + b2 = h2. It follows that

(sin x)2

+ (cos x)2

=

a2 + b2 h2

=

h2 h2

=

1

For convenience, (sin x)2 and (cos x)2 are written more briey as sin2 x and cos2 x respectively. Therefore

we have veri...ed that if x is a given angle then

sin2 x + cos2 x = 1

You are expected to have this identity on your ...nger tips.

Two useful identities are derived from sin2 x + cos2 x = 1 by dividing both sides of the identity by sin2 x or cos2 x.

If we divide both sides of sin2 x + cos2 x = 1 by cos2 x, the result is

sin2 x cos2 x

1

cos2 x + cos2 x = cos2 x OR

sin x 2

12

+1=

cos x

cos x

Since sin x = tan x and 1 = sec x, it follows that (tan x)2 + 1 = (sec x)2. For convenience, we write

cos x

cos x

(tan x)2 and (sec x)2 more briey as tan2 x and sec2 x. Therefore we have veri...ed that

tan2 x + 1 = sec2 x

1

If we divide both sides of sin2 x + cos2 x = 1 by sin2 x, the result is

sin2 x cos2 x

1

cos x 2

12

sin2 x + sin2 x = sin2 x OR 1 + sin x

= sin x

cos x

Since

= cot x and

1

= csc x, it follows that 1 + (cot x)2 = (csc x)2. For convenience, we write

sin x

sin x

(cot x)2 and (csc x)2 more briey as cot2 x and csc2 x. Therefore we have veri...ed that

cot2 x + 1 = csc2 x

You will be required to verify identities. One way to do so is to start with one side of the identity and through a series of algebraic operations, derive the other side of the identity. Here is an example:

sin2 x

Example 1 To verify the identity

+ cos x = sec x.

cos x

We start with the left hand side because there is more we can do with it than the right hand side. Denote the left hand side by LHS and the right hand side by RHS. Then

sin2 x

LHS =

+ cos x

cos x

sin2 x cos2 x

=

+

(since we need a common denominator in order to add the two fractions)

cos x cos x

=

sin2 x + cos2 x =

1

(since sin2 x + cos2 x = 1)

cos x

cos x

= sec x = RHS

We have veri...ed that both sides of the identity are equal. This veri...es the identity.

Exercise 2 1. The trigonometric expression

1

1

+

1 sin x 1 + sin x

can be reduced to:

(A) 2 sin x

(B) 2 cos x

(C) 2 cos2 x

(D) 2 sec2 x

2. Verify each identity:

cos x sec x

a)

= tan x

cot x

c) tan x + cot x = sec x csc x

sec2 x

e)

= sec x csc x

tan x

g) sin x + cos x cot x = csc x

csc2 x

i)

= sec x csc x

cot x

k)

cos2 x sin2 x 1 tan2 x

= cos2 x

b) tan x cos x = sin x d) sec2 x 1 sin2 x = 1

f ) cos x + sin x tan x = sec x

h) sec x cos x = tan x sin x

j) sin2 x 1 + cot2 x = 1

1 cos2 x

l)

= sin x tan x

cos x

2

m) sec x cos x = tan x sin x

tan x cot x o) sin x + sin x = sec x csc x

sin x

cos x

q) 1 cot x tan x 1 = sin x + cos x

cos2 x

s) 1

= sin x

1 + sin x

cos x

u) tan x +

= sec x

1 + sin x

w) cos x + 1 sin x = 2 sec x 1 sin x cos x

y) sin y

cos y = sin y + tan y

1 cot y tan y 1

n) sin x + cos x = sin x + cos x tan x cot x

sin x + cos x cos x sin x

p) sin x

cos x = sec x csc x

tan x r) 1 + tan2 x = sin x cos x

sin2 x

t) 1

= cos x

1 + cos x

v) sec2 x + csc2 x = sec2 x csc2 x

x) sin x + cos x 1 = 0 cos x + 1 sin x

z) (3 cos x 4 sin x)2 + (4 cos x + 3 sin x)2 = 25

3. Write cos4 x sin4 x as a di? erence of two squares, factor it and use the result to show that cos4 x sin4 x = cos2 x sin2 x.

Identities for di?erences or sums of angles

In this section we walk you through a derivation of the identity for cos(y x). This will be used to derive others involving sums or di?erences of angles. Contrary to what one would expect, it is NOT TRUE that cos(y x) = cos y cos x for all angles x and y. Almost any pair of angles x and y gives cos(y x) 6= cos y cos x. For example, the choice y = 120 and x = 30 gives cos (y x) = cos (90 ) = 0 which is not equal to cos 120 cos 30 , (you can easily check that).

To derive the identity, start with a circle of radius 1 and center (0; 0), labelled P in Figure 1 below.

Figure 1 3

Consider a line P Q that makes an angle y, in the second quadrant, with the positive horizontal axis.

Figure 2

Can you see why the coordinates of Q must be (cos y; sin y)? The ...gure below includes a line P R making an angle x in the ...rst quadrant, with the positive horizontal

axis.

Figure 3

The coordinates of R are (cos x; sin x). A standard notation for the length of the line segment QR is jjQRjj. Use the distance formula to show that jjQRjj2 simpli...es to

jjQRjj2 = 2 2 (cos y cos x + sin y sin x)

(1)

If you rotate the circle in Figure 3 counter-clockwise until the ray P R merges into the positive horizontal

4

axis the result is Figure 4 below.

Figure 4

Angle QP R is y x, therefore Q has coordinates (cos (y c) ; sin (y x)). Clearly R has coordinates (1; 0). It follows that the length of QR is also given by

jjQRjj2 = [cos(y x) 1]2 + [sin (y 1)]2

Show that this simpli...es to

jjQRjj2 = 2 2 cos(y x)

(2)

Comparing (1) and (2) reveals that

2 2 cos(y x) = 2 2 (cos y cos x + sin y sin x)

What do you conclude about cos(y x)? You should obtain cos (y x) = cos y cos x + sin y sin x

This is our ...rst identity for the cosine of the di?erence of two angles.

Example 3 We know the exact values of cos 30 , sin 30 , cos 45 and sin 45 . We may use them to calculate

cos 15 . The result: cos 15 = cos (45

pp p 30 ) = cos 45 cos 30 + sin 45 sin 30 = 2 3 + 2 1

2 2 22

pp p p p

= 2 3 + 2 = 6+ 2

4

4

4

Example 4 We are given that x is an angle in the ...rst quadrant with cos x = 1 , y is an angle in the third 4

quadrant with cos y = 1 and we have to determine cos (y x). Since cos (y x) = cos y cos x + sin y sin x, 3

we have to ...nd sin x and sin y. Diagrams will help. In Figure 5 we have a right triangle with a hypotenuse of length 4 and a horizontal side of length 1 because

horizontal coordinate 1 cos x = length of hypotenuse = 4

5

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