Question

√ 1+x - √ 1-x
tan⁻¹ [————————— ]= π/4 - ½ cos⁻¹ x, -1/√2 ≤ x ≤ 1 √ 1+x + √ 1-x

Answer 1

Given   :  tan⁻¹  ( (√1+ x -√ 1- x)/(√1+x +√ 1- x)) = π/4   - ½ cos⁻¹ x  x∈(0, π/4)  -1/√2 ≤ x ≤ 1  

To find :   सिद्ध कीजिए

Solution:

 tan⁻¹  ( (√1+ x -√ 1- x)/(√1+x +√ 1- x)) = π/4  - ½ cos⁻¹ x

LHS =  tan⁻¹  ( (√1+ x - √ 1- x)/(√1+x -√ 1- x))

(√1+ x -√ 1- x)/(√1+ x +√ 1- x))

=  (√1+ x - √ 1- x) (√1+ x +√ 1- x)/(√1+ x +√ 1- x))(√1+ x +√ 1- x))

= ( 1 +  x -(1 -  x) /( 1 +  x + 1 - x  + 2√(1 - x² )  )

=  ( 2x)/(2  + 2√(1 - x² )  )

= x/(1 + √(1 - x² )  )

माना    x = Sin2α

= Sin2α/( 1 + Cos2α)

Cos2α =  Cos²α - Sin²α

= Sin2α/( 1 + Cos²α - Sin²α)

1 - Sin²α =Cos²α

= Sin2α/( Cos²α +  Cos²α)

= 2SinαCosα/2Cos²α

= Sinα/Cosα

= tanα

LHS =  tan⁻¹  ( tanα)

= α

 x = Sin2α

=>x = Cos (π/2 - 2α)

=> Cos⁻¹x  = π/2 - 2α

=> 2α =  π/2 - Cos⁻¹x

=> α =   (1/2) (π/2 - Cos⁻¹x)

=> α = π/4  - (1/2)Cos⁻¹x

LHS = α

= π/4  - (1/2)Cos⁻¹x

= RHS

LHS = RHS

=>   tan⁻¹  ( (√1+ x -√ 1- x)/(√1+x +√ 1- x)) = π/4   - ½ cos⁻¹ x

QED

इति सिद्धम

और सीखें

tan⁻¹(2/11) + tan⁻¹ (7/24) = tan⁻¹ ½

brainly.in/question/16554897

3cos⁻¹ x = cos⁻¹ (4x³ - 3x), x∈[1/2, 1 ] को सिद्ध कीजिये

brainly.in/question/16554602

"cos⁻¹(-1/2 ) का मुख्य मान ज्ञात कीजिये"

brainly.in/question/16549994

cos⁻¹ (½) + 2sin⁻¹(½) का मान ज्ञात कीजिये

brainly.in/question/16554597

2sin⁻¹35 = tan⁻¹ 247 सिद्ध कीजिए

https://brainly.in/question/16556013