CHAPTER VII. ON THE RELATIONS BETWEEN THE TRIGONOMETRICAL RATIOS OF THE SAME ANGLE. 102. THE following relations are evident from the definitions: 105. Euclid I. 47 gives us that in any right-angled triangle the square on the hypotenuse the sum of the squares on the perpendicular and on the base, = or, (hypotenuse)3 = (perpendicular)2 + (base)3. (i) Divide each side of this identity by (ii) Divide each side of the same identity by (base), and we get (iii) Divide each side of the same identity by are each a statement in Trigonometrical language of Euc. I. 47. 107. We give the above proof in a different form. To prove that cos2 0 + sin2 0 = 1. Let ROE be any angle 0. R M In OE take any point P, and draw PM perpendicular to OR. Then with respect to 0, MP is the perpendicular, OP is the hypotenuse, and OM is the base; 2 We have to prove that sin2 + cos2 0 = 1, i.e. that MP2 + OM2 = OP2. But this is true by Euclid I. 47. Therefore cos2 + sin2 0 = 1. Similarly we may prove that and that 1 + tan2 0 = sec3 0, 1+ cot2 = cosec2 0. 108. The following is a LIST OF FORMULE with which the student must make himself familiar: 109. In proving Trigonometrical identities it is often convenient to express the other Trigonometrical Ratios in terms of the sine and cosine. Example. Prove that tan A+cot A=sec A. cosec A. and this is true, because sin2 A + cos2 A=1. 110. Sometimes it is more convenient to express all the other Trigonometrical Ratios in terms of the sine only, or in terms of the cosine only. Example. Prove that sin10+2 sin2 . cos2 = 1 - cos1 0. Hence, putting 1 - cos2 6, and (1 - cos2 6)2 for sin2 and sin1 0 respectively, we have to prove that or that or that (1 − cos2 0)2+2. (1 − cos2 ). cos2 = 1 − cos1 0, 1-2 cos2 + cos10+2 cos2 0-2 cos1 0-1-cos1 0, which is true. 1- cos10=1- cos1 0, This example may be proved directly, by reversing the steps of the above proof; thus ... (1-2 cos2 + cos1 0) + 2 cos2 - 2 cos1 0=1 - cos1 0, .: (1-cos2 )2+2 cos2 (1 - cos2 ) = 1 - cos1 0, .. (sin20)2+2 cos20. sin2 = 1-cos1. Q.E.D. NOTE. (1-cos 6) is called the versed sine of ; it is |