cos⁻¹ 45+ cos⁻¹ 1213= cos-1 3365 सिद्ध कीजिए
Answers
Given : Cos⁻¹ 4/5 +cos⁻¹ 12/13 = Cos⁻¹ 33/65
To find : सिद्ध कीजिए
Solution:
Cos⁻¹ 4/5 +cos⁻¹ 12/13 = Cos⁻¹ 33/65
माना x = Cos⁻¹ 4/5
=>Cosx = 4/5
sin²x = (1 - cos²x)
=> sin²x = ( 1 - (4/5)²)
=> sin²x = (3/5)²
=> sinx = 3/5
=> x = sin⁻¹(3/5)
y = Cos⁻¹ 12/13
=>Cosy =12/13
sin²y = (1 - cos²y)
=> sin²y = ( 1 - (12/13)²)
=> sin²y = (5/13)²
=> siny = 5/13
y = sin⁻¹(5/13)
Cos⁻¹ 4/5 +cos⁻¹ 12/13 = Cos⁻¹ 33/65
दोनों तरफ cos लेने पर
cos (Cos⁻¹ 4/5 +cos⁻¹ 12/13) = cos (Cos⁻¹ 33/65)
LHS = cos (Cos⁻¹ 4/5 +cos⁻¹ 12/13)
Cos(x + y) = CosxCosy - SinxSiny)
= Cos(Cos⁻¹ 4/5)Cos(cos⁻¹ 12/13) - Sin(sin⁻¹(3/5))Sin(sin⁻¹(5/13))
= (4/5)(12/13) - (3/5)(5/13)
= 48/65 - 15/65
= 33/65
RHS = cos (Cos⁻¹ (33/65))
=33/65
LHS = RHS
=>Cos⁻¹ 4/5 +cos⁻¹ 12/13 = Cos⁻¹ 33/65
QED
इति सिद्धम
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