18. Show that the sum of the tangent and cotangent of an angle is equal to the product of the secant and cosecant of the angle. Recalling the values given on page 8, find the value of x when: cos x)2 - 1 = (sin x cos x)2+1, find x.' 23. 3 tan2 24. tang+cot 31. Given (sin x 36. Given sin x = 4 cos r, find sinx and cos x. 38. From the formula tan x = under which tan x = sin x. sin x , find the condition Solve the following equations; that is, find the value of x when : 49. sin A+cos A = (1+tan 4) cos 4. 51. cos x : cot x = 50. cot x COS X = √1+ cot2x. 1 √1-cos2r. cos2x Find the values of the other functions of A when : 73. Given cot 22° 30′ = √2+1, find the other functions of 22° 30' 74. Write tan2 A+ cot2 4 so as to contain only cos A. In the triangle ABC, prove the following relations: 75. sin Asin (B+ C). 76. 'cos A 77. tan A 78. cot A cos (2 A + B + C'). == cos (B+ C). tan (B+ C). 83. sin Acos (A+B+C'). cot (B+ C'). 79. sin Asin (2A+B+C). 80. sin B C: : 86. sin (A+B) = cos (B-C) 87. sin (C-4)=-cos (B+C) sin (A+2B+C). 88. cos Bcos (A +2 B+ C). 81. cos Ccos (1+B+20). 89. tan Atan (2 A + B + C). 82. cot B cot (4+2 B+C'). =cot 90. cot A tan (B+C+A). In the quadrilateral ABCD, prove the following relations: CHAPTER VI FUNCTIONS OF THE SUM OR THE DIFFERENCE OF TWO ANGLES 90. Formula for sin (x + y). In this figure there are shown two acute angles, x and y, with ZAOC acute and equal to x + y; two perpendiculars are let fall from C, and two from D, as shown. Then by geometry the triangles CGD and EOD are similar and hence. <GCD= LEOD = x. Considering the radius as unity, OD = cos y and CD sin y. Hence we have Hence sin (x + y) = sin x cos y + cos x sin y. This is one of the most important formulas and should be memorized. 91. Formula for cos (x + y). Using the above figure we see that cos (x + y) = OF : OE- DG. Hence cos (x + y) = = cos x cos y — sin x sin y. This important formula should be memorized. sin 45° sin 45° G D x B 92. The Proofs continued. In the proofs given on page 97, x, y, and x + y were assumed to be acute angles. If, however, x and y are acute but x+y is obtuse, as shown in this figure, the proofs remain, word for word, the same as before, the only difference being that the sign of OF will be negative, as DG is now greater than OE. This, however, does not affect the proof. The above formulas, therefore, hold true for all acute angles x and y. Furthermore, if these formulas hold true for any two acute angles x and y, they hold true when one of the angles is increased by 90°. Thus, if for a we write x' = 90° + x, then, by § 87, But by § 87, and Α' F O E Hence by substituting these values, sin(x' + y) = sin x' cos y + cos x' sin y. That is, § 90 holds true if either angle is repeatedly increased by 90°. It is therefore true for all angles. Similarly, by § 87, cos (x' + y) = cos (90° + x + y) = − sin(x + y) sin x cos y - cos x sin y = cos x' cos y sin x' sin y, -- by substituting cos x' for - sin x and sin a' for cos x as above. That is, § 91 also holds true if either angle is repeatedly increased by 90°. It is therefore true for all angles. Exercise 41. Sines and Cosines 2 Given sin 30° = cos 60° = 1, cos 30° = sin 60° = √3, and sin 45° = cos 45° = }}√2, find the values of the following: Dividing each term of the numerator and denominator of the last of these fractions by cos x cos y, we have cot (x + y): == = sin(x + y) sin x cos y + cos x sin y whatever the size of the angles x and y (§ 92). Dividing each term of the numerator and denominator of the last of these fractions by sina sin y, and then remembering that |