1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. EXAMPLES XIII. √3+tan 40° +tan 80° = √√3. tan 40°. tan 80°. cos A cos (B+ C) – cos B cos (A + C) = sin (A – B) sin C. sin A cos (B+ C) – sin B cos (A + C) = sin (4 – B) cos C. If A + B + C is an odd multiple of π sin2 B + sin2 C = sin2 A + 2 cos A sin B sin C. cos (a + B) + sin (a − ẞ) = 2 sin (‡π + a) cos (†π + ß). sin a + sin ẞ + sin y − sin (a + B + y) = 4 sin 1⁄2 (ẞ + y) sin 1⁄2 (y + a) sin § (a + ß). cos a + cos ẞ + cos y + cos (a + B + y) = 4 cos (B+y) cos 1 (y + a) cos 1 (a + B). sin a + sin ß − sin y − sin (a + ß − y) sin (ẞ + y − a) + sin (y + a − ß) + sin (a + ß − y) - sin (a+B+y)= 4 sin a sin ẞ sin y. cos (B+ y− a) + cos (y + a − ẞ) + cos (a + B − y) + cos (a + B + y) = 4 cos a cos ẞ cos y. cos2 x + cos2 y + cos2 z + cos2 (x + y + z) = 2 {1 + cos (y + z) cos (z+x) cos (x + y)}. sin2 x + sin2 y + sin2 z + sin2 (x + y + z) = 2 {1 — cos (y + z) cos (≈ + x) cos (x + y)}. 18. sin a sin (ẞ− y) + sin ẞ sin (y − a) + sin y sin (a – ẞ) = 0. 19. sin (a+ẞ+y+d) = cos a cos ẞ cos y cos d (tan a + tan ß + tan y + tan 8) 20. sin a sin ẞ sin y sin 8 (cot a + cot ẞ + cot γ cos (SA) cos (S-B) cos (S-C) cos (S— D) + sin (S-A) sin (S-B) sin (S-C) sin (SD) if =cos A cos B cos C cos D+ sin A sin B sin C sin D 2S = A + B + C + D. 21. If a right-angle is divided into three parts whose tangents are lp, mp, np; 26. 27. = 2 tan A sec2 B 2 tan B sec2 A 1- tan2 A tan2 B* 2 cot A cosec2 B 2 cot B cosec2 A cot2 B-cot2 A 2 sec A sec B sec (A + B) + sec (A – B) = sec2 4 + sec2 B – sec2 A sec2 B 2 cox A cox B sec (A + B) – sec (A – B) = cox2 A cox2 B — cox2 A — cox2 B If A, B, C are the angles of a triangle, prove 28-33: sin A sin B sin C sin A cos B cos C + sin B cos C cos A+ sin C cos A cos B. 1 + cos A cos B cos C = cos A sin B sin C + cos B sin C sin A + cos C sin A sin B. tan A tan B tan C tan A+ tan B+ tan C. = 31. cot B cot C + cot C cot A + cot A cot B=1. 32. cos A cos B cos Csin A sin B cos C + sin B sin C cos A+ sin C sin 4 cos B. 33. tan B tan C + tan C tan A+ tan A tan B=1. 34. If A, B, C, D are the angles of a quadrilateral, (i) cot A + cot B + cot C + cot D=tan A tan B tan C'tan D, (ii) sin A sin B cos C cos D + ... = cos A cos B cos C cos D + sin A sin B sin C sin D – 1. 35. If 0, 4, are the angles which any straight line, in the plane of the triangle ABC, makes with BC, CA, AB respectively, then then 36. 37. then (i) a2 sin (Þ +4 − 0) + b2 sin († + 0 − p) + c2 sin (0 + 0 − 4) +2bc sin 0 + 2ca sin & + 2ab sin ↓ = 0. (ii) a2 cos (+ 4 − 0) + b2 cos (4 + 0 − p) + c2 cos (0 + − ¥) +2bc cos 0 + 2ca cos &+2ab cos &= 0. If sin sin a sin (ø + a − ß) = sin ø sin (a + ß) sin (a + 0), sin & sin a sin (0+ a + B) = sin 0 sin (a – ẞ) sin (a + ¢). If cos e cos a cos (+ a − ẞ) = cos o sin (a + ẞ) sin (a + 0), cos & cos a cos (0+ a + ẞ) = cos 0 sin (a – ẞ) sin (a + ). 38. In a triangle right-angled at C, and if ẞ and y are unequal, then each member and cot 0 = 43. sin (ẞ + y) sin (y + a) sin (a + ß) cos (B+y) cos (y + a) cos (a + ß) + sin2 (a + ß + y) * If cos (a + B + y) − sin (a + B + y) then either a or ẞ or y is of the form (n − 1)π. 2 sin 0 sec3 0 1- tan2 a tan2 (' sin (A + B) = sin A cos B + cos A sin B ...... 2.69 .(1), ..(2), (3). .(4). ...(5). .(6). 353. Now we may observe that (cos A + sin A)2 = cos2 A + 2 sin A cos A + sin2 A, Comparing these expansions with the above formulæ (4), (5), (6), we see that and first term - last term of (cos A + sin A)2 = cos 2A, middle term of (cos A + sin A)2 = sin 24, first term – last term of (1 + tan A)2 = den. of tan 24, middle term of (1 + tan 4)2 = num. of tan 24. |