Prove that:tan -1 (1 + x/ 1 – x) = π/4 + tan -1 x, x < 1

1.	Prove that:tan -1 (1 + x/ 1 – x) = π/4 + tan -1 x, x < 1
Answer» tan ^-1 (1 + x/ 1 – x) = π/4 + tan ^-1 x, x < 1 Consider x = tan θ Where θ = tan ^-1 x Here LHS = tan ^-1 (1 + x/ 1 – x) By substituting the value = tan ^-1 (1 + tan θ/ 1 – tan θ) = tan ^-1 (tan [π/4 + θ]) So we get = π/4 + θ By substituting the value of θ = π/4 + tan ^-1 x = RHS Hence, it is proved.

Answer»

tan ^-1 (1 + x/ 1 – x) = π/4 + tan ^-1 x, x < 1

Consider x = tan θ

Where θ = tan ^-1 x

Here LHS = tan ^-1 (1 + x/ 1 – x)

By substituting the value

= tan ^-1 (1 + tan θ/ 1 – tan θ)

= tan ^-1 (tan [π/4 + θ])

So we get

= π/4 + θ

By substituting the value of θ

= π/4 + tan ^-1 x

= RHS

Hence, it is proved.

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