For the first three parts of this question, we will examine different ways of doing the integral: 1 = sin5 x dx. = sin, (a) By using the substitution u = cosx, show that: 2 - 1 = cosx+cos 3 Cos x - - Cos x + C [4 marks] (b) Find the power reduction formula for sin 5 x. Use this to show that: 5 1 5 cos 5x + cos 3x - - cosx+C 80 48 8 [6 marks] (c) Use De Moivre's theorem to show that: cos 5x = 16 cos5 x 20 cos x + 5 cosx. Then use this to verify that the expressions in (a) and (b) are the same. You may also use the identity cos 3x = 4 cos x 3 cos x without proving it. [6 marks] The last two parts of this question are on the vectors a = 31+ 6j + 2k and b = 8i + 4j+k. (d) What is the angle between a and b? Give your answer in radians to 4 significant digits. (e) Find a unit vector which is perpendicular to both a and b. [5 marks] [4 marks]