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5-4 Sum and Difference Identities Write each trigonometric expression as an algebraic expression. 25. SOLUTION: If we let and , then . Sketch one right triangle with an acute and angle and another with an acute angle . Label the sides such that and . Then use the Pythagorean Theorem to express the length of each third side. and Using these triangles, we find that these values in . Substitute and simplify. 27. SOLUTION: If we let angle and , then and another with an acute angle and . Label the sides such that Pythagorean Theorem to express the length of each third side. eSolutions Manual - Powered by Cognero . Sketch one right triangle with an acute and . Then use the Page 1 5-4 Sum and Difference Identities 27. SOLUTION: If we let angle and , then and another with an acute angle and . Sketch one right triangle with an acute . Label the sides such that and . Then use the Pythagorean Theorem to express the length of each third side. Using these triangles, we find that . Substitute these values in , , and and simplify. 29. SOLUTION: eSolutions Manual - Powered by Cognero Page 2 5-4 Sum and Difference Identities 29. SOLUTION: If we let , then . Sketch a right triangle with an acute angle . Label the sides such that . Then use the Pythagorean Theorem to express the length of the third side. Using this triangle, we find that . Substitute this value in and simplify. eSolutions Manual - Powered by Cognero Page 3 triangle, we find that 5-4 Using Sum this and Difference Identities . Substitute this value in and simplify. 31. SOLUTION: eSolutions Manual - Powered by Cognero If we let , then . Sketch a right triangle with an acute angle . Label the sides such that . Then use the Pythagorean Theorem to express the length of the third side. Page 4 5-4 Sum and Difference Identities 31. SOLUTION: If we let , then . Sketch a right triangle with an acute angle . Label the sides such that . Then use the Pythagorean Theorem to express the length of the third side. Using this triangle, we find that . Substitute this value in and simplify. Verify each cofunction identity using one or more difference identities. 33. sec = csc x eSolutions Manual - Powered by Cognero SOLUTION: Page 5 5-4 Sum and Difference Identities Verify each cofunction identity using one or more difference identities. 33. sec = csc x SOLUTION: Verify each reduction identity. – ) = – cos 35. cos ( SOLUTION: Find the solution to each expression on the interval [0, 2π). 41. cos( + x) + cos ( + x) = 1 SOLUTION: On the interval [0, 2 ), eSolutions Powered by Cognero 43. sin Manual - + sin = SOLUTION: when x = and x = . Page 6 the and interval [0, 2 ), when x = 5-4 On Sum Difference Identities 43. sin + sin and x = . = SOLUTION: On the interval [0, 2 ), 45. tan ( + x) + tan ( when x = and x = . + x) = 2 SOLUTION: On the interval [0, 2 ), when x = and x = . Verify each identity. 47. cot – tan = SOLUTION: 49. 2 sinManual a cos- bPowered = sin (a + b) + sin eSolutions by Cognero SOLUTION: (a – b) Page 7 5-4 Sum and Difference Identities 49. 2 sin a cos b = sin (a + b) + sin (a – b) SOLUTION: eSolutions Manual - Powered by Cognero Page 8