tan⁻¹ 6316 = sin⁻¹ 513 +cos⁻¹ 35 सिद्ध कीजिए
Answers
Given : tan⁻¹ 63/16 = sin⁻¹ 5/13 +cos⁻¹ 3/5
To find : सिद्ध कीजिए
Solution:
tan⁻¹ 63/16 = sin⁻¹ 5/13 +cos⁻¹ 3/5
माना x = sin⁻¹ 5/13
=>Sinx =5/13
Cos²x = (1 - Sin²x)
=> Cos²x = ( 1 - (5/13)²)
=> Cos²x = (12/13)²
=> Cosx = 12/13
sinx/cosx = (5/13)(12/13)
=> tanx = 5/12
=> x = tan⁻¹(5/12)
माना y = Cos⁻¹ 3/5
=>Cosy = 3/5
sins²y = (1 - cos²y)
=>sin²y = ( 1 - (3/5)²)
=> sin²y = (4/5)²
=> siny = 4/5
=> siny/cosy = (4/5)/(3/5)
=> tany = 4/3
=> y = tan⁻¹(4/3)
tan⁻¹ 63/16 = sin⁻¹ 5/13 +cos⁻¹ 3/5
=> tan⁻¹ 63/16 = tan⁻¹(5/12) + tan⁻¹(4/3)
दोनों तरफ Tan लेने पर
= tan( tan⁻¹ 63/16) = tan( tan⁻¹(5/12) + tan⁻¹(4/3))
LHS = tan( tan⁻¹ 63/16)
= 63/16
RHS = tan( tan⁻¹(5/12) + tan⁻¹(4/3))
tan(x + y) = (tanx + tany)/(1 - tanxtany)
= ((5/12)+(4/3))/ (1 - (5/12)(4/3) )
= (15 + 48)/(36 - 20)
= 63/16
LHS = RHS
=>tan⁻¹ 63/16 = sin⁻¹ 5/13 +cos⁻¹ 3/5
QED
इति सिद्धम
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