Math, asked by utkggjduwa5028, 11 months ago

सिद्ध कीजिए
tan⁻¹x-- √=12cos⁻¹ (1-x1+x), x∈ [0,1]

Answers

Answered by amitnrw
0

Given   :  tan⁻¹ (√x)  = (1/2) Cos⁻¹ (( 1-x)/(1 + x )  x∈ [0,1]

To find :   सिद्ध कीजिए

Solution:

 tan⁻¹ (√x)  = (1/2) Cos⁻¹ (( 1-x)/(1 + x )

RHS = (1/2) Cos⁻¹ (( 1-x)/(1 + x )

माना x = tan²α    => tanα  = √x  => α = tan⁻¹(√x)

= (1/2) Cos⁻¹ (( 1-tan²α)/(1 + tan²α ))

=  (1/2) Cos⁻¹ (( 1-sin²α/cos²α)/(1 +sin²α/cos²α ))

=  (1/2) Cos⁻¹ (( cos²α -sin²α )/(cos²α +sin²α  ))

cos²α -sin²α = Cos2α

cos²α +sin²α  = 1

= (1/2)  Cos⁻¹  ( Cos2α)

= (1/2) 2α

= α

= tan⁻¹(√x)

= LHS

LHS = RHS

=>  tan⁻¹ (√x)  = (1/2) Cos⁻¹ (( 1-x)/(1 + x )

QED

इति सिद्धम

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