Evaluate each of the following:(i) tan-1(tan 2)(ii) tan-1(tan 4)(iii)

1.	Evaluate each of the following:(i) tan-1(tan 2)(ii) tan-1(tan 4)(iii) tan-1(tan 12)
Answer» (i) Given as tan^-1(tan 2) As tan^-1(tan x) = x if x ϵ [-π/2, π/2] Here x = 2 which does not belongs to above range We have tan (π – θ) = –tan (θ) So, tan (θ – π) = tan (θ) tan (2 – π) = tan (2) Now 2 – π is in the given range Hence, tan^–1 (tan 2) = 2 – π (ii) Given as tan^-1(tan 4) As tan^-1(tan x) = x if x ϵ [-π/2, π/2] But here x = 4 which does not belongs to above range We also have tan (π – θ) = –tan (θ) So, tan (θ – π) = tan (θ) tan (4 – π) = tan (4) Now 4 – π is in the given range Thus, tan^–1 (tan 2) = 4 – π (iii) Given as tan^-1(tan 12) As tan^-1(tan x) = x if x ϵ [-π/2, π/2] Here x = 12 which does not belongs to above range As we know that tan (nπ – θ) = –tan (θ) tan (θ – 2nπ) = tan (θ) Here n = 4 tan(12 – 4π) = tan (12) Now 12 – 4π is in the given range Therefore, tan^–1(tan 12) = 12 – 4π.

Evaluate each of the following:(i) tan-1(tan 2)(ii) tan-1(tan 4)(iii) tan-1(tan 12)

Answer»

(i) Given as tan^-1(tan 2)

As tan^-1(tan x) = x if x ϵ [-π/2, π/2]

Here x = 2 which does not belongs to above range

We have tan (π – θ) = –tan (θ)

So, tan (θ – π) = tan (θ)

tan (2 – π) = tan (2)

Now 2 – π is in the given range

Hence, tan^–1 (tan 2) = 2 – π

(ii) Given as tan^-1(tan 4)

As tan^-1(tan x) = x if x ϵ [-π/2, π/2]

But here x = 4 which does not belongs to above range

We also have tan (π – θ) = –tan (θ)

So, tan (θ – π) = tan (θ)

tan (4 – π) = tan (4)

Now 4 – π is in the given range

Thus, tan^–1 (tan 2) = 4 – π

(iii) Given as tan^-1(tan 12)

As tan^-1(tan x) = x if x ϵ [-π/2, π/2]

Here x = 12 which does not belongs to above range

As we know that tan (nπ – θ) = –tan (θ)

tan (θ – 2nπ) = tan (θ)

Here n = 4

tan(12 – 4π) = tan (12)

Now 12 – 4π is in the given range

Therefore, tan^–1(tan 12) = 12 – 4π.