The value of \(tan^{-1}\left(tan \frac{3\pi}{4}\right)\)isA. \(\frac{

1.	The value of \(tan^{-1}\left(tan \frac{3\pi}{4}\right)\)isA. \(\frac{3\pi}{4}\)B. \(\frac{\pi}{4}\)C. \(\frac{-\pi}{4}\)D. none of these
Answer» Correct Answer is \(\frac{-\pi}{4}\) Now, let x = \(tan^{-1}\left(tan \frac{3\pi}{4}\right)\) ⇒ tan x =tan ( \(\frac{3\pi}{4}\)) Here range of principle value of tan is [\(-\frac{\pi}{2},\frac{\pi}{2}\) ] ⇒ x = \(\frac{3\pi}{4}\)∉ [\(-\frac{\pi}{2},\frac{\pi}{2}\)] Hence for all values of x in range [\(-\frac{\pi}{2},\frac{\pi}{2}\)] ,the value of \(tan^{-1}\left(tan \frac{3\pi}{4}\right)\) is ⇒ tan x =tan (π - \(\frac{\pi}{4}\)) (\(\because\) tan (\(\frac{3\pi}{4}\))= tan ( π - \(\frac{\pi}{4}\)) ) ⇒ tan x =tan (\(\frac{-\pi}{4}\)) (\(\because\) tan (π - θ)= tan(-θ)) ⇒ x =\(\frac{-\pi}{4}\)

The value of \(tan^{-1}\left(tan \frac{3\pi}{4}\right)\)isA. \(\frac{3\pi}{4}\)B. \(\frac{\pi}{4}\)C. \(\frac{-\pi}{4}\)D. none of these

Answer»

Correct Answer is \(\frac{-\pi}{4}\)

Now, let x = \(tan^{-1}\left(tan \frac{3\pi}{4}\right)\)

⇒ tan x =tan ( \(\frac{3\pi}{4}\))

Here range of principle value of tan is [\(-\frac{\pi}{2},\frac{\pi}{2}\) ]

⇒ x = \(\frac{3\pi}{4}\)∉ [\(-\frac{\pi}{2},\frac{\pi}{2}\)]

Hence for all values of x in range [\(-\frac{\pi}{2},\frac{\pi}{2}\)] ,the value of

\(tan^{-1}\left(tan \frac{3\pi}{4}\right)\) is

⇒ tan x =tan (π - \(\frac{\pi}{4}\)) (\(\because\) tan (\(\frac{3\pi}{4}\))= tan ( π - \(\frac{\pi}{4}\)) )

⇒ tan x =tan (\(\frac{-\pi}{4}\)) (\(\because\) tan (π - θ)= tan(-θ))

⇒ x =\(\frac{-\pi}{4}\)