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Please solve RD Sharma class 12 chapter Inverse Trignometric Functions exercise 3.7 question 3 sub question (vii) maths textbook solution

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Answer: 4-\pi

Hint: The range of principal value of  \tan ^{-1} is \left [ -\frac{\pi }{2} ,\frac{\pi }{2}\right ]
Given:  \tan ^{-1}(\tan 4)


As       \tan ^{-1}(\tan x)=x \text { if } x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] 

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

            \begin{aligned} &\tan (\pi-\theta)=-\tan (\theta) \\ &\tan (\theta-\pi)=\tan \theta \\ &\tan (4-\pi)=\tan (4) \end{aligned}

Now,     4-\pi \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]

Hence,  \tan ^{-1}(\tan 4)=4-\pi

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