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TEST 4 - REVIEW Verify the identiry. 1) tan2 x = sec2 x - sin2 x - cos2 x Answer: Identity 2) sin x sin x + = 2 csc x 1 - cos x 1 + cos x Answer: Identity 3) cos2 x - sin2 x = 1 - 2 sin2 x Answer: Identity 4) cot2 x 1 + sin x = csc x - 1 sin x Answer: Identity 5) 1 + csc x = cos x + cot x sec x Answer: Identity 6) sin3 x + sin x cos2 x = cos x tan x Answer: sin3 x + sin x cos2 x sin x(sin2 x + cos2 x) = tan x tan x = sin x · 1 tan x Pythagorean identity = sin x ÷ sin x cos x = sin x · cos x sin x = cos x Prove the identity. 7) sec3 x cos x - sec2 x sin2 x = 1 Answer: sec3 x cos x - sec2 x sin2 x = = 1 cos3 x 1 cos2 x - · cos x - 1 · sin2 x 2 cos x sin2 x cos2 x = sec2 x - tan2 x =1 Pythagorean identity 1 Find the exact value using a double-angle identity. 10 8) sin 3 3 2 Answer: - Use an identity to write the expression as a single trigonometric function or as a single number. 2 tan 15° 9) 1 - tan2 15° Answer: 3 3 10) 2 cos2 22.5° - 1 Answer: 2 2 Find the exact value by using a half-angle identity. 11) sin 22.5° Answer: 1 2 2- 2 Simplify the expression by using a sum or difference identity. 12) cos(14k) cos (4k) - sin(14k) sin (4k) Answer: cos(18k) Use trigonometric identities to find the exact value. tan 43° + tan 17° 13) 1 - tan 43° tan 17° Answer: 3 14) sin 10° cos 50° + cos 10° sin 50° Answer: 3 2 Use an appropriate identity to find the exact value of the expression. 15) cos (75°) Answer: 6- 2 4 Find the exact value by using a sum or difference identity. 16) tan 105° Answer: -2 - 3 2 Find the exact value of the expression using the provided information. 12 15 , with A in quadrant II, and sin B = , with B in quadrant II. 17) Find cos(A - B) given that cos A = 13 17 Answer: 171 221 Find the exact value of the composition. 1 18) cos arcsin 4 Answer: 15 4 Evaluate the expression exactly. 19) csc(csc-1 3) Answer: 3 Solve the equation exactly over the interval [0, 2 ). 1 20) sin x cos x = 2 Answer: 5 4 4 , Find all real numbers in the interval [0, 2 ) that satisfy the equation. 21) sin2 x - cos2 x = 0 Answer: 3 5 7 , , 4 4 4 4 , Find all real numbers in [0, 2 ] that satisfy the equation. 3 22) sin 4x = 2 Answer: 2 7 7 13 5 19 , , , , , 12 6 3 12 6 12 3 12 , , Find all values of x in the interval [0°, 360°) that satisfy the equation. Round approximate answers to the nearest tenth of a degree. tan 4x - tan x 3 = 23) 1 + tan 4x tan x 3 Answer: {10°, 70°, 130°, 190°, 250°, 310°} 24) 7 cot2 x - 5 = 0 Answer: {49.8°, 130.2°, 229.8°, 310.2°} Find all values of in [0°, 360°) that satisfy the equation. 25) 2 sin 2 - 3 = 0 Answer: {30°, 60°, 210°, 240°} 3