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5. (a) Find the expansion of cos(x + y) and cos(x – y). Show that cos(x + y) cos(x - y)
be written as cos x - sin?
y.
(5 marks)
(b) Hence, from part (a) and without using the calculator, find the value of
cos(75°) cos (15°), where x and y are special angles.
(5 marks)

Respuesta :

Answer:

The identity is: cos(x+y)cos(x-y) = cos^2(x) - sin^2(y).  Answer to part (b) is 1/4

Step-by-step explanation:

cos(x+y) = cos(x) cos(y) - sin(x)sin(y)

cos(x-y) = cos(x)cos(y) + sin(x)sin(y)

For convenience, let A = cos(x) cos(y), and B = sin(x) sin(y).

Thus,

cos(x+y) = A - B

cos(x-y) = A + B, and

cos(x+y) cos(x-y) = (A-B)(A+B) = A^2 - B^2 = cos^2(x) cos^2(y) - sin^2(x)sin^2(y) = (1 - sin^2(x))(1 - sin^2(y)) - sin^2(x) sin^2(y) =

1 - sin^2(x) - sin^2(y) + sin^2(x) sin^2(y) - sin^2(x) sin^2(y) =

1 - sin^2(x) - sin^2(y) =

cos^2(x) - sin^2(y)

(b) Let x=45, and y=30, so we have that

cos(75)cos(15) = cos(45+30) cos(45-30) =

cos^2(45) - sin^2(30) =

(sqrt(2)/2)^2 - (1/2)^2 =

2/4 - 1/4 =

1/4

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