1. Verify a trigonometric identity
EXAMPLE 4
Verify a trigonometric identity
Verify the identity = sin q .
sec q – 1 2
sec q
sec q – 1 2
sec q =
sec q 2 – 1
Write as separate fractions.
= 1 – ( ) 1
sec q 2
Simplify.
= 1 – cos q 2
Reciprocal Identity
= sin q 2
Pythagorean Identity

2. Verify a trigonometric identity
EXAMPLE 5
Verify a trigonometric identity
Verify the identity sec x + tan x =
cos x
1 – sin x 1
cos x
+ tan x =
sec x + tan x
Reciprocal Identity 1
cos x +
sin x =
Tangent Identity
1 + sin x
cos x =
Add fractions.
1 + sin x
cos x =
1 – sin x
Multiply by
1 – sin x

3. Verify a trigonometric identity
EXAMPLE 5
Verify a trigonometric identity
1 – sin x
cos x (1 – sin x) = 2
Simplify numerator.
cos x
cos x (1 – sin x) = 2
Pythagorean Identity
cos x
1 – sin x =
Simplify.

4. EXAMPLE 6
Verify a real-life trigonometric identity
Shadow Length
A vertical gnomon (the part of a sundial that projects a shadow) has height h. The length s of the shadow cast by the gnomon when the angle of the sun above the horizon is q can be modeled by the equation below. Show that the equation is equivalent to s = h cot q .
h sin (90° – q )
sin q = s

5. Verify a real-life trigonometric identity
EXAMPLE 6
Verify a real-life trigonometric identity
SOLUTION
Simplify the equation.
h sin (90° – q )
sin q = s
Write original equation.
h sin ( – q )
sin q π 2 =
Convert 90° to radians.
h cos q
sin q =
Cofunction Identity
= h cot q
Cotangent Identity

6. GUIDED PRACTICE
for Examples 4, 5, and 6
Verify the identity.
6. cot (–q ) = –cot q
SOLUTION
cot (–q ) =
tan (–θ ) 1
–tan ( θ ) 1 =
= –cot θ

7. GUIDED PRACTICE
for Examples 4, 5, and 6
7. csc2 x (1 – sin2 x) = cot2 x
SOLUTION 1
sin2 x
= cos2 x
csc2 x (1 – sin2 x )
= cot2 x

8. GUIDED PRACTICE
for Examples 4, 5, and 6
8. cos x csc x tan x = 1
SOLUTION
cos x csc x tan x
= cos x csc x
sin x
cos x
= cos x 1
sin x
cos x
= 1

9. GUIDED PRACTICE
for Examples 4, 5, and 6
9. (tan2 x + 1)(cos2 x – 1) = – tan2 x
SOLUTION
(tan2 x + 1)(cos2 x – 1)
= sec2 x
(–sin2x) 1
cos2 x
(–sin2x) =
= –tan2 x