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

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

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.

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

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

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

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

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

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