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  • Please solve RD Sharma Class 12 Chapter Inverse Trigonometric Functions Exercise 3.3 Question 1 Subquestion (iii) maths textbook solution.

Please solve RD Sharma Class 12 Chapter Inverse Trigonometric Functions Exercise 3.3 Question 1 Subquestion (iii) maths textbook solution.

Answers (1)

Answer : 0

Hint : The branch with range \left[\frac{-\pi}{2}, \frac{\pi}{2}\right] is called the principal value branch of function \tan^{-1}.

           \tan ^{-1}: R \rightarrow\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]

Given : \tan ^{-1}\left(\cos \frac{\pi}{2}\right)

Explanation :

Let  y=\tan ^{-1}\left(\cos \frac{\pi}{2}\right)                                                                                                  ....(i)

y=\tan ^{-1}(0) \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\because \cos \frac{\pi}{2}=0\right]

\tan \; y=0

\tan y=\tan 0^{\circ} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\because \tan 0^{\circ}=0\right]

y=0

\tan ^{-1}\left(\cos \frac{\pi}{2}\right)=0

The range of principal value branch of \tan^{-1} is \left[\frac{-\pi}{2}, \frac{\pi}{2}\right].

\because \quad \tan ^{-1}\left(\cos \frac{\pi}{2}\right)=0 \in\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]

Hence, the principal value of \tan ^{-1}\left(\cos \frac{\pi}{2}\right) is 0.

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