Hi, Can you please help me with these maths questions by showing me the answer with steps and a small explanation, I only need help with questions 14 to 18 I have completed the others:

Question 2 De Moivre Theorem Write the expressions for Throughout this SAC, you will be using the following equations: a) sin(0 - 4) cis(0) = cos(0) + isin(0) ... ... ... ... .. (1) cis(0) x cis(4) = cis(0 + 4 ) .......... (2) b) cos(0 (cis(@) )" = cis(ne ) . . . . . . .. (3) where i = v-1 and n is a positive integer and all angls are in radians. In addition, you will need to use the following binomial expansions: (A + B) 3 = A3 + 3A2B + 3AB2 + B3 (A + B) 4 = A4 + 4A3B + 6A2B2 + 4AB3 + B4 Question 1 Use equations (1) and (2) to prove the following addition formulas: sin (0 + 4) = sin(0) cos(4) + cos(0)sin(q) cos(0 + 4) = cos(0) cos(4) - sin(0)sin(4) (5 marks) NQuestion 2 Write the expressions for a) sin(0 - 4) (1 mark) b) cos(8 - 4) (1 mark) Question 3 Use the results of Question 2 with suitable values of 0 & 4 to find the exact values of the following. Show your working and give answers with rational denominators. a) sin (2) (3 marks) b) cos () (3 marks) c) tan () (3 marks) 3Question 4 Question 5 Use equations (1) an Use the results of Question 1 with suitable values of 0 & 4 to find the exact values of the following Show your working and give answers with rational denominators. a) sin (") (3 marks) b) cos (3 marks) c) tan () (3 marks)wing Question 5 Use equations (1) and (3) with n = 2 to prove the following double-angle formulas: 3 marks) sin(20) = 2sin(0) cos (0) cos(20) = cos2 (0) - sin2(0) (4 marks) Question 6 Use the results of Question 5 to prove the following. Show your working. a) cos(20) = 2cos2(0) - 1 (2 marks) b) cos(20) = 1 - 2sin2(0) (2 marks)Use the results of Question 7 Question 7 Question 8 Show your working and Use the results of Question 6 to prove the following. Show your working. sin a) a) sin(0) = + 1-cos (20) (2 marks) b) cos(0) = + 1+cos (20) 2 (2 marks) 6Question 8 Use the results of Question 7 to find the exact values of the following. Show your working and give answers with rational denominators. marks) a) sin (2) (3 marks) b) cos (3 marks) c) tan (3 marks)Question 9 Question 10 Compare the results of Questions 3 & 8. Use equations a) Show that the two expressions for sin ( ) are equal. (2 man b) Show that the two expressions for cos () are equal. (2 marks) 8Question 10 (2 marks) Use equations (1) and (3) with n = 3 to prove the following formulas: cos(30) = cos3 (0) - 3sin2(0)cos(0) sin(30) = 3sin(0) cos2(0) - sin3 (0) (6 marks) Question 11 Use the results in Question 10 to find: a) Expression for cos(30) in terms of cos(0) only. (4 marks) b) Expression for sin(30) in terms of sin(0) only. (4 marks) 9Question 13 a) Use the resu Question 12 a) Use the result in Question 11a to prove by substitution that x = cos ( ) is a solution of the equation. 8x3 - 6x - 1 = 0 (4 marks) b) Use CAS to find the solutions of the equation in part a (to 5 decimal places) and hence, confirm that one of these solutions is indeed x = cos (5). (4 marks) 10cos is a solution of the equation: Question 13 Use the result in Question 11b to prove by substitution that x = sin ( ) is a solution of the equation. (4 marks) 8x3 - 6x + 1 = 0 (4 marks) b) Use CAS to find the solutions of the equation in part a (to 5 decimal places) and hence, confirm that one of these solutions is indeed x = sin (18). (4 marks) 11Question 16 a) Use the result i Question 14 Use equations (1) and (3) with n = 4 to prove the following formulas: cos(40) = cos*(0) - 6sin?(0)cos2(0) + sin*(0) sin(40) = 4sin(0) cos3(0) - 4sin3(0) cos(0) (8 marks) Question 15 Show that the first result of Question 5 can also be written as follows: cos(40) = 8 cos*(0) - 8 cos (0) + 1 (4 marks) 12Question 16 (8 marks) Use the result in Question 6 to prove by substitution that x = 2cos ( ) is a solution of the equation: x4 - 4x2 + 1 =0 (4 marks) b) Use CAS to find the exact solutions of the equation in part a (to 5 decimal places) and hence, confirm that one of these solutions is indeed x = 2cos (2) (4 marks) Total mark = 100 End of SAC 3 13