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Please Solve RD Sharma Class 12 Chapter Inverse Trigonometric Function Exercise 3.10 Question 1 Subquestion (iii) Maths Textbook Solution.

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Answer: 1
Hint: Check if there is any relation between \sec ^{-1}x  and \cos^{-1}x . Use inverse trigonometric functions properties to solve. ( \therefore \tan^{-1}\frac{1}{x} = \cot^{-1}x )
Given:\sin\left (\tan^{-1}x+ \tan^{-1}\frac{1}{x} \right )\: f\! or\: x> 0
Solution: Let replace\tan^{-1}\frac{1}{x}  by( \cot^{-1}x) , x < 0
Now,\Rightarrow \sin\left (\tan^{-1}x+ \tan^{-1}\frac{1}{x} \right )
\Rightarrow \sin\left (\tan^{-1}x+ \cot^{-1}x \right )
Again by property,\tan^{-1}x + \cot^{-1}x= \frac{\pi }{2}
= \sin\left ( \frac{\pi }{2} \right )
= \sin\frac{\pi }{2}
= 1
Concept: Properties and relations between inverse trigonometric functions.
Note: Inverse trigonometric functions remember relation between all trigonometric functions. Also, try to remember value of trigonometric functions.  

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