(tan a+sec a)" - (tan a-seca)" (103) The sum of n terms of the series log (1-2 cos 0) + log (1 − 2 cos 20) + log (1 − 2 cos 22 ) + etc. (107) Solve the equation 3+x2 − 2x − 1 = 0. [Result. 2 cos π, 2 cos π, 2 cos.] (108) (x-cosπ) (x − 2 cos ‡π) (x − 2 cos §π) (x − 2 cos §π) (109) The sum to n terms of the series 12 cos 2a +22 cos 4a+32 cos 6a+...to n terms хо = 4 1+2h cos x + 2h2 cos 2x + etc. α -b (a + 1 sin2 0 - 1 (a + b)2 b απ b\3 2 sin2 20 (115) 1+ + + = }} {e≈ + 2e− 1× cos § (x √3)}. 3! 6! ... (116) The roots of the equation x" sin na-nx”-1 sin (na+B) + n (n-1) x-2 sin (na+2ß) — ... = 0 2! are given by x=sin (a+ß−kp) cosec (0−kp) where k has all integral values from 0 to n−1 and no=π. (117) Find the general value of ✪ which satisfies the equation (cos + i sin ) (cos 20+ i sin 20) ... (cos ne + i sin n✪) = 1. (118) When n is even and if np=Ã, tan a tan (a+6) tan (a +24) ..... tan {a + (n− 1) p}=(− 1)3. where the sum of the mth powers of the root of equation z2-1=0. = (121) If e π = 2"+1 2" cos e cos 20 cos 22 0...cos 2"-10=1. (122) If a, b, c, are the roots of the equation then ... 2-1 x2 − P1x2−1+ P2x2—2 – р3x2¬3 + etc., tan−1a+tan1b+tan-1c+...=tan-1 P1-P3+P5 − (123) Prove that 1-P2+P3-.. e"+e"=2(1+22) {1+(3)} {1+()}... (124) The sum of the products of the reciprocals of the fourth powers of every positive integer is 3848 5!9! y = 2 (127) Prove that the coefficients of 62 and 4 in the expression vanish; explaining à priori why they do so. (128) Having given the formula 2202 2202 0088-(1-2) (1-325)... cos = deduce the expression for sin 0 in factors. 2 (129) The coefficient of " in the expansion of (1 + x) (1 + 1) (1 + 1)......... 22 T2n is (2n+1)!' (130) By putting aia for in the expression of sin in (131) If a series of points are distributed symmetrically round the circumference of a circle, the sum of the squares of their distances from a point on the circumference is twice that from the centre. (132) If A1, A2, A3, ... A2n+1 are angular points of a regular polygon inscribed in a circle and O any point in the circumference between A1 and A2n+19 then the sum of the lengths = -the sum OA2+OA1+ОÃ ̧+...+0A2n• (133) If from a point P straight lines PB1, PB2,...PB2 be drawn to the middle points of the sides of a closed polygon A1A... A, and if the angles PB1A1, PB,A, ... PВ„А„ be denoted by a1, ag,... a, respectively, and the triangles PA12, PA2A3, ... PA„A1 by A1, A2, An, prove that EXAMINATION PAPERS. I. SANDHURST-FURTHER. Nov. 1882. 1. Name and define the trigonometrical ratios. Prove that sec2 A+cosec2 A = sec2 A cosec2 A. If the cosecant of an angle between 90° and 180° is what is the √3 secant? And if the cosine of an angle between 540° and 630° is -, what is the cosecant? 3. In a plane triangle ABC prove that— i. tan A tan B tan C-tan A+ tan B+tan C. ii. a sin A+ b sin B+c sin C-2 (a cos A+ẞ cos B+ycos C), where abc are the sides and aßy the perpendiculars let fall on them from the opposite angles respectively. and if R, r are the radii of the circumscribing and inscribed circles |