cos⁻¹ 1213 + sin⁻¹ 35 = sin⁻¹ 5665 सिद्ध कीजिए
Answers
Given : Cos⁻¹ 12/13 +sin⁻¹ 3/5 = sin⁻¹ 56/65
To find : सिद्ध कीजिए
Solution:
Cos⁻¹ 12/13 +sin⁻¹ 3/5 = sin⁻¹ 56/65
माना x = Cos⁻¹ 12/13
=>Cosx =12/13
sin²x = (1 - cos²x)
=> sin²x = ( 1 - (12/13)²)
=> sin²xy = (5/13)²
=> sinx = 5/13
x = sin⁻¹(5/13)
माना y = sin⁻¹ 3/5
=>Siny = 3/5
cos²y = (1 - sin²y)
=> cos²y = ( 1 - (3/5)²)
=> cos²y = (4/5)²
=> cosy = 4/5
=> y = cos⁻¹(4/5)
Cos⁻¹ 12/13 +sin⁻¹ 3/5 = sin⁻¹ 56/65
दोनों तरफ Sin लेने पर
Sin (Cos⁻¹ 12/13 +sin⁻¹ 3/5) =Sin (sin⁻¹ 56/65)
LHS = Sin (Cos⁻¹ 12/13 +sin⁻¹ 3/5)
Sin(x + y) = SinxCosy + CosxSiny
= Sin(sin⁻¹(5/13))Cos(cos⁻¹ 4/5)+ Cos(Cos⁻¹ 12/13)Sin(sin⁻¹(3/5))
= (5/13)(4/5) + (12/13)(3/5)
= 20/65 + 36/65
= 56/65
RHS = Sin(Sin⁻¹ (33/65))
=56/65
LHS = RHS
=>Cos⁻¹ 12/13 +sin⁻¹ 3/5 = sin⁻¹ 56/65
QED
इति सिद्धम
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