Two statements are given below. For each, an erroneous proof is provided. There is one main error in each proof. State what this error is and explain why it is an error. (Note that the rst of these statements is true and the second is false, but this is irrelevant to the question and your answer.) (a) Statement: For all real numbers a such that cost 75 0 and sina: 75 0, tanmicotrr, = SBCZBCSCIL'. Proof: We are aiming to prove that tanm + cotx = secsccsc .12. Using the facts that sin m cos a: 1 1 , tan :1: = , cot x = _ , sec :13 = , and csc a: = _ , we obtaln that cos a: sin x cos x sin a: sin :5 + cos :1: 1 1 cos .r sin :17 cos a: sin .r sin2 a: + cos2 a: 1 cos a: sin 23 cos a: sin m 1 1 cos a: sin .1: cos a: sin a: The last line results from applying the Pythagorean Identity. The last equality is clearly true and so the statement is proved. |:l (b) Statement: All perfect squares are positive. Proof: Let n be an integer. 2 Case I: n, is positive. If n, is positive, then H, is positive because it is the product of two positive integers. Case 2: n is negative. If n is negative, then 11,2 is positive because it is the product of two negative integers. |:|