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Questions discussed in this lecture

Solve:
9. $\tan ^{-1} (x-1)+\tan ^{-1} x+\tan ^{-1} (x+1)=\tan ^{-1} 3x$

10. $3\sin ^{-1} \frac{2x}{1+x^{2} } -4\cos ^{-1} \frac{1-x^{2} }{1+x^{2} } +2\tan ^{-1} \frac{2x}{1-x^{2} } =\frac{\pi }{3}$

11. If $\sin ^{-1} \frac{2a}{1+a^{2} } -\cos ^{-1} \frac{1-b^{2} }{1+b^{2} } =\tan ^{-1} \frac{2x}{1-x^{2} }$, then prove that $x=\frac{a-b}{1+ab}.$

12. Evaluate: $\tan ^{-1} \left(\frac{a+bx}{b-ax} \right),\, \, x<\frac{b}{a}$

13. Prove: $\tan ^{-1} \left(\frac{a-b}{1+ab} \right)+\tan ^{-1} \left(\frac{b-c}{1+bc} \right)+\tan ^{-1} \left(\frac{c-a}{1+ca} \right)=0$

14. If $\tan ^{-1} x+\tan ^{-1} y=\frac{4\pi }{5}$, then find the value of $\cot ^{-1} x+\cot ^{-1}y$?

15. If $\tan ^{-1} \left(\frac{1}{1+1.2} \right)+\tan ^{-1} \left(\frac{1}{1+2.3} \right)+…+\tan ^{-1} \left(\frac{1}{1+n.(n+1)} \right)=\tan ^{-1} \phi$, then find the value of $\phi$.

16. If $(\tan ^{-1} x)^{2} +(\cot ^{-1} x)^{2} =\frac{5\pi ^{2} }{8}$, then find x.