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