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Please prove the following: 2cos -1 (3/root 13) + cot -1 (16/63) + 1/2cos -1 (7/25) = pi cos[2tan -1 (1/7)] = sin[4tan -1 (1/3)]

Please prove the following:
  1. 2cos-1 (3/root 13) + cot-1(16/63) + 1/2cos-1(7/25) = pi
  2. cos[2tan-1(1/7)] = sin[4tan-1(1/3)]

Grade:12

1 Answers

viswanath
40 Points
9 years ago
   2cos-1(3/√13) +cot-1(16/63) +1/2cos-1(7/25)
=2tan-1(2/3) +tan-1(63/16) + 1/2tan-1(24/7)
=tan-1(12/5) + tan-1(63/16) + 1/2(2tan-1(3/4))
=tan-1(12/5) + tan-1(63/16) + tan-1(3/4)
= (tan-1(12/5) + tan-1(3/4)) + tan-1(63/16)
= (pi-tan-1(63/16)) +tan-1(63/16)
= pi
 
cos(2tan-1(1/7))= cos(tan-1(7/24))= cos(cos-1(24/25))=24/25
sin(4tan-1(1/3))=sin(2tan-1(3/4))=sin(tan-1(24/7))=sin(sin-1(24/25))=24/25

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