Ex 2.2, 5

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Ex 2.2, 5 Write the function in the simplest form: tan-1 (√(1 + x^2 ) − 1)/x , x ≠ 0 tan-1 (√(1 + x^2 ) − 1)/x Putting x = tan 𝜃 = tan-1 ((√(1 + tan^2 θ )− 1)/(tan θ)) = tan-1 ((√(sec^2 θ ) − 1)/(tan θ)) = tan-1 ((sec⁡θ − 1)/(tan θ)) = tan-1 ((1/cos⁡𝜃 − 1)/(sin⁡𝜃/cos⁡𝜃 )) = tan-1 (((1 − cos⁡θ)/cos⁡θ )/(sin⁡𝜃/cos⁡𝜃 )) = tan-1 ((1 −〖 cos〗⁡𝜃)/cos⁡𝜃 ×cos⁡𝜃/sin⁡𝜃 ) = tan-1 ((1 −〖 cos〗⁡𝜃)/sin⁡𝜃 ) = tan-1 ((2 𝑠𝑖𝑛2 𝜃/2)/(2 〖sin 〗⁡〖𝜃/2〗 cos⁡〖 𝜃/2〗 )) = tan-1 (〖sin 〗⁡〖𝜃/2〗/cos⁡〖 𝜃/2〗 ) = tan-1 (𝑡𝑎𝑛 𝜃/2) = θ/2 We assumed that x = tan θ θ = tan-1x Hence, tan-1 (√(1 + x^2 ) − 1)/x = θ/2 = 1/2 tan-1x