show that 2 Tan inverse 5 / 12= pie / 4 tan inverse 1/ 70 - tan inverse 1 /99
Answers
Answer:
show that 2 Tan inverse 5 / 12= pie / 4 tan inverse 1/ 70 - tan inverse 1 /99
Answer:
Step-by-step explanation:
4 tan-1(1/5) – tan-1(1/70) + tan-1(1/99)
= 2[2 tan-1(1/5)] – [tan-1(1/70) – tan-1(1/99)]
Using the formulas 2 tan-1(x) = tan-1[2x/(1 – x2)] and tan-1(x) – tan-1(y) = tan-1[(x – y)/(1 + xy)],
= 2 tan-1[2(1/5)/ {1 – (1/5)2}] – tan-1[{(1/70) – (1/99)}/ {1 + (1/70)(1/99)}]
= 2 tan-1[10/(25 – 1)] – tan-1[(99 – 70)/(6930 + 1)]
= 2 tan-1(10/24) – tan-1(29/6931)
= 2 tan-1(5/12) – tan-1(29/6931)
= tan-1[2(5/12)/ {1 – (5/12)2} – tan-1(1/239)
= tan-1[120/ (144 – 25)] – tan-1(1/239)
= tan-1(120/119) – tan-1(1/239)
= tan-1[{(120/119) – (1/239)}/ {1 + (120/119)(1/239)}]
= tan-1[(120 × 239 – 119)/ (239 × 119 + 120)]
= tan-1(28561/ 28561)
= tan-1(1)
= π/4