## Inverse Trigonometric Functions Lecture 5 Part 2

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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.

## HOTS Questions Inverse Trigonometry Lecture 4 Part 3

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#### Clip 1

Questions discussed in this lecture

Prove:
21. $\tan ^{-1} \frac{63}{16} =\sin ^{-1} \frac{5}{13} +\cos ^{-1} \frac{3}{5}$
(NCERT Miscellaneous Exercise Q7)

22. $\cos ^{-1} \frac{4}{5} +\cos ^{-1} \frac{12}{13} =\cos ^{-1} \frac{33}{65}$
(NCERT Miscellaneous Exercise Q5)

23. $\cos ^{-1} \frac{12}{13} +\sin ^{-1} \frac{3}{5} =\sin ^{-1} \frac{56}{65}$
(NCERT Miscellaneous Exercise Q6)

24. $\sin ^{-1} \frac{12}{13} +\cos ^{-1} \frac{4}{5} +\tan ^{-1} \frac{63}{16} =\pi$
(NCERT Example 11)

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#### Clip 2

25. $2\tan ^{-1} \left\{\tan \frac{\alpha }{2} \tan \left(\frac{\pi }{4} -\frac{\beta }{2} \right)\right\}=\tan ^{-1} \frac{\sin \alpha \cos \beta }{\cos \alpha +\sin \beta }$

26. $2\tan ^{-1} \left(\sqrt{\frac{a-b}{a+b} } \tan \frac{\theta }{2} \right)=\cos ^{-1} \left(\frac{a\cos \theta +b}{a+b\cos \theta } \right)$

27. If $\sin \left(\sin ^{-1} \frac{1}{5} +\cos ^{-1} x\right)=1$, then find the value of x.
(NCERT Exercise 2.2 Q14)

28. If $y=\cot ^{-1} (\sqrt{\cos x} )-\tan ^{-1} (\sqrt{\cos x} )$, prove that $\sin y=\tan ^{2} \frac{x}{2}$

## Practice Questions Inverse Trigonometry Lecture 4 Part 2

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#### Clip 1

Questions discussed in this lecture

Prove:
11. $\frac{9\pi }{8} -\frac{9}{4} \sin ^{-1} \frac{1}{3} =\frac{9}{4} \sin ^{-1} \frac{2\sqrt{2} }{3}$
(NCERT Miscellaneous Exercise Q12)

12. $\tan \left(\frac{\pi }{4} +\frac{1}{2} \cos ^{-1} \frac{a}{b} \right)+\tan \left(\frac{\pi }{4} -\frac{1}{2} \cos ^{-1} \frac{a}{b} \right)=\frac{2b}{a}$

13. $\sin [\cot ^{-1} \{ \cos (\tan ^{-1} x)\} ]=\frac{\sqrt{x^{2} +1} }{\sqrt{x^{2} +2} }$

14. $\tan ^{-1} \left(\frac{x}{\sqrt{a^{2} -x^{2} } } \right)=\sin ^{-1} \frac{x}{a} =\cot ^{-1} \left(\frac{\sqrt{a^{2} -x^{2} } }{a} \right)$

15. $\tan ^{-1} \left(\frac{m}{n} \right)-\tan ^{-1} \left(\frac{m-n}{m+n} \right)=\frac{\pi }{4}$

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#### Clip 2

16. $\tan ^{-1} \frac{1}{5} +\tan ^{-1} \frac{1}{7} +\tan ^{-1} \frac{1}{3} +\tan ^{-1} \frac{1}{8} =\frac{\pi }{4}$
(NCERT Miscellaneous Exercise Q8)

17. $4\tan ^{-1} \frac{1}{5} -\tan ^{-1} \frac{1}{70} +\tan ^{-1} \frac{1}{99} =\frac{\pi }{4}$

18. $\tan ^{-1} \left(\frac{\cos x}{1-\sin x} \right)-\cot ^{-1} \left(\sqrt{\frac{1+\cos x}{1-\cos x} } \right)=\frac{\pi }{4}$

19. $\sin ^{-1} \frac{3}{5} -\sin ^{-1} \frac{8}{17} =\cos ^{-1} \frac{84}{85}$
(NCERT Example 10)

20. $\sin ^{-1} \frac{8}{17} +\sin ^{-1} \frac{3}{5} =\tan ^{-1} \frac{77}{36}$
(NCERT Miscellaneous Exercise Q4)

## HOTS Question Inverse Trigonometric Functions L3 P3

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Questions discussed in this lecture

Simplify:
15. $\tan ^{-1} \sqrt{\frac{a-x}{a+x} }$

16. $\tan ^{-1} \left(\frac{\cos x}{1-\sin x} \right),\, \, -\frac{\pi }{2} (NCERT Example 5)

17. $\sin ^{-1} \left\{\frac{x+\sqrt{1-x^{2} } }{\sqrt{2} } \right\}$

18. $\tan ^{-1} (x+\sqrt{1+x^{2} } )$

19. $\sin \left\{2\tan ^{-1} \sqrt{\frac{1-x}{1+x} } \right\}$

20. $\sin ^{-1} \left(\frac{\sin x+\cos x}{\sqrt{2} } \right)$

21. $\tan ^{-1} \left[\frac{a\cos x-b\sin x}{b\cos x+a\sin x} \right],\, {\rm if}\, \, \frac{a}{b} \tan x>-1$ (NCERT Example 12)

22. $\cot ^{-1} \left(\frac{\sqrt{1+\sin x} +\sqrt{1-\sin x} }{\sqrt{1+\sin x} -\sqrt{1-\sin x} } \right)=\frac{x}{2} ,\, \, x\in \left(0,\, \, \frac{\pi }{4} \right)$ (NCERT Miscellaneous Exercise Q10)

23. $\tan ^{-1} \left(\frac{\sqrt{1+x} -\sqrt{1-x} }{\sqrt{1+x} +\sqrt{1-x} } \right)=\frac{\pi }{4} -\frac{1}{2} \cos ^{-1} x,\, \, -\frac{1}{\sqrt{2} } \le x\le 1$ (NCERT Miscellaneous Exercise Q11)

## Practice Questions Inverse Trigonometry Lecture 3 Part 2

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Questions discussed in this lecture

Simplify:
6. $\tan ^{-1} \frac{x}{\sqrt{a^{2} -x^{2} } } ,\, \, |x| (NCERT Exercise 2.2 Q9)

7. $\tan ^{-1} \left(\frac{3a^{2} x-x^{3} }{a^{3} -3ax^{2} } \right),\, a>0;\, \frac{-a}{\sqrt{3} } \le x\le \frac{a}{\sqrt{3} }$ (NCERT Exercise 2.2 Q10)

8. $\tan ^{-1} \left(\frac{\cos x}{1+\sin x} \right)$

9. $\tan ^{-1} \left(\frac{\sin x}{1+\cos x} \right)$

10. $\tan ^{-1} \left(\frac{a\cos x-b\sin x}{b\cos x+a\sin x} \right)$

11. $\tan ^{-1} \left(\frac{x}{a+\sqrt{a^{2} -x^{2} } } \right)$

12. $\sin ^{-1} \left(\frac{5}{13} \cos x+\frac{12}{13} \sin x\right)$

13. $\sin ^{-1} (x\sqrt{1-x} -\sqrt{x} \sqrt{1-x^{2} } )$

14. $\sin ^{-1} \left\{\frac{\sqrt{1+x} +\sqrt{1-x} }{2} \right\}$

## HOTS questions inverse trigonometry lecture 2 part 4

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Evaluate each of the following:

19. $\tan \left\{2\tan ^{-1} \frac{1}{5} -\frac{\pi }{4} \right\}$

20.$\sin ^{-1} \left(\cos \left(\frac{43\pi }{5} \right)\right)$

24. $\sec ^{2} (\tan ^{-1} 2)+{\rm cosec}^{2} (\cot ^{-1} 3)$

29. $\cos (\tan ^{-1} x)$

30. $\sin ^{-1} \frac{1}{2} -2\sin ^{-1} \frac{1}{\sqrt{2} }$

31. $\tan \left(\cos ^{-1} \frac{4}{5} +\tan ^{-1} \frac{2}{3} \right)$ (NCERT Exercise 2.2 Q18)

32. $\tan \left(2\tan ^{-1} \frac{1}{5} \right)$

33. $\tan \left[2\cos \left(2\sin ^{-1} \frac{1}{2} \right)\right]$ (NCERT Exercise 2.2 Q11)

34. $\cot (\tan ^{-1} a+\cot ^{-1} a)$ (NCERT Exercise 2.2 Q12)

36. $\sin (\tan ^{-1} x),\, \, |x|<1$ (NCERT Miscellaneous Exercise Q15)

37. $\tan ^{-1} \left(\frac{x}{y} \right)-\tan ^{-1} \left(\frac{x-y}{x+y} \right)$ (NCERT Miscellaneous Exercise Q17)

## Derivation of identities for inverse trigo functions part 3

#### Clip 1

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Summary of this lecture:
Derivation of identities for Inverse Trigonometry Functions

How to study Inverse Trigonometry from my notes and video lectures

Evaluate each of the following:

15. $\tan \left[\frac{1}{2} \cos ^{-1} \left(\frac{2}{\sqrt{5} } \right)\right]$

21. $\sin ^{-1} \left(-\frac{\sqrt{3} }{2} \right)+\cos ^{-1} \left(-\frac{1}{2} \right)+\tan ^{-1} \left(-\frac{1}{\sqrt{3} } \right)$

22. $\tan ^{2} (\sec ^{-1} 2)+\cot ^{2} ({\rm cosec}^{-1} 3)$

23. $\sin \left(2\tan ^{-1} \frac{1}{3} \right)+\cos (\tan ^{-1} 2\sqrt{2} )$

#### Clip 2

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25. $\sin (\tan ^{-1} x+\cot ^{-1} x)$

26. $\sin \left(\cos ^{-1} \frac{4}{5} \right)$

27. $\sin \left(\cot ^{-1} \frac{4}{3} \right)$

28. $\sin (\cot ^{-1} x)$

35. $\tan \frac{1}{2} \left[\sin ^{-1} \frac{2x}{1+x^{2} } +\cos ^{-1} \frac{1-y^{2} }{1+y^{2} } \right],\, \, |x|<1,\, y>0\, {\rm and}\, xy<1$ (NCERT Exercise 2.2 Q13)

## Twenty practice questions inverse trigo lecture 2 part 2

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Summary of this lecture:
Revision of Ranges of Inverse Trigonometric Functions

20 Practice Questions from my booklet

Evaluate each of the following:

1. $\sin ^{-1} \left(-\frac{\sqrt{3} }{2} \right)$

2. $\cot ^{-1} \left(\frac{-1}{\sqrt{3} } \right)$ (NCERT Example 2)

3. $\tan ^{-1} \left(-\frac{1}{\sqrt{3} } \right)$

4. $\tan ^{-1}(1)+\cos ^{-1} \left(-\frac{1}{2} \right)+\sin ^{-1} \left(-\frac{1}{2} \right)$

5. $\cos ^{-1} \left(\frac{1}{2} \right)+2\sin ^{-1} \left(\frac{1}{2} \right)$ (NCERT Exercise 2.2 Q12)

6. $\tan ^{-1} \sqrt{3} -\sec ^{-1} (-2)$ (NCERT Exercise 2.2 Q14)

7. $\sin ^{-1} \left(\sin \frac{4\pi }{5} \right)$

8. $\sin ^{-1} \left(\sin \frac{2\pi }{3} \right)$ (NCERT Exercise 2.2 Q16)

9. $\tan ^{-1} \left(\tan \frac{3\pi }{4} \right)$ (NCERT Exercise 2.2 Q17)

10. $\cos ^{-1} \left(\cos \frac{7\pi }{6} \right)$ (NCERT Exercise 2.2 Q19)

11. $\sin \left(\frac{\pi }{3} -\sin ^{-1} \left(-\frac{1}{2} \right)\right)$ (NCERT Exercise 2.2 Q20)

12. $\tan ^{-1} \sqrt{3} -\cot ^{-1} (-\sqrt{3} )$ (NCERT Exercise 2.2 Q21)

13. $\csc ^{-1} (-2)$

14. $\sin ^{-1} \left(\sin \frac{3\pi }{5} \right)$ (NCERT Example 9)

16. $\cos ^{-1} \left(\cos \frac{13\pi }{6} \right)$ (NCERT Miscellaneous Exercise Q1)

17. $\tan ^{-1} \left(\tan \frac{7\pi }{6} \right)$ (NCERT Miscellaneous Exercise Q2)

18. $\cos ^{-1} [\cos (-680{}^\circ )]$

20. $\tan ^{-1} \left(\tan \frac{5\pi }{6} \right)$

## HOTS Questions Inverse Trigonometric Functions Lecture 5

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Questions discussed in this lecture

Solve:
1. $\tan ^{-1} \frac{x-1}{x-2} +\tan ^{-1} \frac{x+1}{x+2} =\frac{\pi }{4}$
(NCERT Exercise 2.2 Q15)

2. $\tan ^{-1} 2x+\tan ^{-1} 3x=\frac{\pi }{4}$
(NCERT Example 13)

3. $\tan ^{-1} (x+1)+\tan ^{-1} (x-1)=\tan ^{-1} \frac{8}{31}$

4. $2\tan ^{-1} (\cos x)=\tan ^{-1} (2\, {\rm cosec}\, x)$
(NCERT Miscellaneous Exercise Q13)

5. $\tan ^{-1} \frac{1-x}{1+x} =\frac{1}{2} \tan ^{-1} x,\, (x>0)$
(NCERT Miscellaneous Exercise Q14)

6. $\sin ^{-1} (1-x)-2\sin ^{-1} x=\frac{\pi }{2}$
(NCERT Miscellaneous Exercise Q16)

7. $\sin ^{-1} x+\sin ^{-1} (1-x)=\cos ^{-1} x$

8. $\sin ^{-1} 6x+\sin ^{-1} 6\sqrt{3} x=-\frac{\pi }{2}$

परिंदों को मंज़िल मिलेगी यकीनन,
ये फैले हुए उनके पर बोलते हैं |
वो लोग रहते हैं ख़ामोश अक्सर,
ज़माने में जिनके हुनर बोलते हैं ||

## Proving Questions Inverse Trigonometry Functions Lecture 4

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Questions discussed in this lecture

Prove:
1. $3\sin ^{-1} x=\sin ^{-1} (3x-4x^{3} ),\, \, x\in \left[-\frac{1}{2} ,\frac{1}{2} \right]$
(NCERT Exercise 2.2 Q1)

2. $3\cos ^{-1} x=\cos ^{-1} (4x^{3} -3x),\, \, x\in \left[\frac{1}{2} ,\, \, 1\right]$
(NCERT Exercise 2.2 Q2)

3. $\tan ^{-1} \frac{2}{11} +\tan ^{-1} \frac{7}{24} =\tan ^{-1} \frac{1}{2}$
(NCERT Exercise 2.2 Q3)

4. $2\tan ^{-1} \frac{1}{2} +\tan ^{-1} \frac{1}{7} =\tan ^{-1} \frac{31}{17}$
(NCERT Exercise 2.2 Q4)

5. $\tan ^{-1} \frac{3}{4} +\tan ^{-1} \frac{3}{5} -\tan ^{-1} \frac{8}{19} =\frac{\pi }{4}$

6. $\cot ^{-1} 7+\cot ^{-1} 8+\cot ^{-1} 18=\cot ^{-1} 3$

7. $\tan ^{-1} x+\tan ^{-1} \left(\frac{2x}{1-x^{2} } \right)=\tan ^{-1} \left(\frac{3x-x^{3} }{1-3x^{2} } \right)$
(NCERT Example 7)

8. $\sin ^{-1} (2x\sqrt{1-x^{2} } )=2\sin ^{-1} x=2\cos ^{-1} x$
(NCERT Example 8, 9)

9. $\tan ^{-1} \left(\frac{\sqrt{1+x^{2} } +\sqrt{1-x^{2} } }{\sqrt{1+x^{2} } -\sqrt{1-x^{2} } } \right)=\frac{{\rm \pi }}{2} -\frac{1}{2} \sin ^{-1} x^{2}$

10. $\tan ^{-1} \sqrt{x} =\frac{1}{2} \cos ^{-1} \left(\frac{1-x}{1+x} \right),\, \, x\in [0,\, 1]$
(NCERT Miscellaneous Exercise Q9)

मुश्किल इस दुनिया में कुछ भी नहीं,
फिर भी लोग अपने इरादे तोड़ देते हैं |
अगर सच्चे दिल से हो चाहत कुछ पाने की,
तो सितारे भी अपनी जगह छोड़ देते हैं ||