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## Answers

where n is any integer.

Now the principal value of tan−1(x) lies in [−π2,π2] precisely in (0,π2)

if finite x>0.

So, the principal value of tan−1(2)+tan−1(3) will lie in (0,π).

So, the principal value of tan−1(2)+tan−1(3) will be 3π4. Interestingly, the principal value of tan−1(−1) is −π4.

But the general values of tan−1(2)+tan−1(3) and tan−1(−1) are same.

Alternatively, tan−1(1)+tan−1(2)+tan−1(3)=tan−1(1+2+3−1⋅2⋅31−1⋅2−2⋅3−3⋅1)=tan−1(0)=mπ, where m is any integer.

Now the principal value of tan−1(1)+tan−1(2)+tan−1(3) will lie in (0,3π2) which is π.

The principal value of tan−1(0) is 0≠π.