Question

Write the values of x for which 2tan^-1x

Answer 1

hello....
We will learn how to prove the property of the inverse trigonometric function, 2 arctan(x) = arctan(2x1−x2) = arcsin(2x1+x2) = arccos(1−x21+x2)

or, 2 tan−1 x = tan−1 (2x1−x2) = sin−1 (2x1+x2) = cos−1(1−x21+x2)

Proof: 

Let, tan−1 x = θ         

Therefore, tan θ = x

We know that,

tan 2θ = 2tanθ1−tan2θ

tan 2θ = 2x1−x2

2θ = tan−1(2x1−x2)

2 tan−1 x = tan−1(2x1−x2) …………………….. (i)

Again, sin 2θ = 2tanθ1+tan2θ

sin 2θ = 2x1+x2

2θ = sin−1(2x1+x2 )

2 tan−1 x = sin−1(2x1+x2) …………………….. (ii)

Now,  cos 2θ = 1−tan2θ1+tan2θ

 cos 2θ =  1−x21+x2

2θ = cos−1 (1−x21+x2)

2 tan−1 x = cos (1−x21+x2) …………………….. (iii)

Therefore, from (i), (ii) and (iii) we get, 2 tan−1 x = tan−1 2x1−x2= sin−1 2x1+x2 = cos−1 1−x21+x2                   Proved

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