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

Answer:$-1$
Hint: Check if there is any relation between$\sec ^{-1}x$  and $\cos^{-1}x$ . Use inverse trigonometric functions properties to solve.$( \therefore \tan^{-1}\frac{1}{x} = - \pi + \cot^{-1}x )$
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  $(- \pi + \cot^{-1}x) , x < 0$
Now,$\sin\left (\tan^{-1}x+ \tan^{-1}\frac{1}{x} \right )$
$\Rightarrow \sin\left (\tan^{-1}x- \pi + \cot^{-1}x \right )$
Again by property,  $\tan^{-1}x+ \cot^{-1}x = \frac{\pi }{2}$
$\! \! \! \! \! \! \! \! \! \! = \sin\frac{\pi }{2}- \pi \\ = \sin-\frac{\pi }{2}\\ = - 1$
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.