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EXAMPLE 4

2. Verify the identity = sin q . 1. –. =. 2. 2. 2. 2. 2. 2. 2. sec q – 1. sec q. sec q. sec q – 1. sec q . sec q. sec q. 2. 1. = 1 – ( ). sec q. 2. = 1 – cos q . 2. = s in q . EXAMPLE 4. Verify a trigonometric identity.

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EXAMPLE 4

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

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

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

  4. Shadow Length A vertical gnomon (the part of a sundial that projects a shadow) has heighth. The length sof 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 = hcotq . h sin (90° – q ) = sinq EXAMPLE 6 Verify a real-life trigonometric identity s

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

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

  7. = cos2x 1 sin2 x for Examples 4, 5, and 6 GUIDED PRACTICE 7.csc2x (1 – sin2x) = cot2x SOLUTION csc2x (1 – sin2x ) = cot2 x

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

  9. (tan2x + 1)(cos2x – 1) 1 cos2x (–sin2x) = sec2x (–sin2x) = for Examples 4, 5, and 6 GUIDED PRACTICE 9. (tan2x + 1)(cos2x – 1) = – tan2x SOLUTION = –tan2x

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