Type – I
$\sin \theta =\frac{1}{{\rm cosec}\, \theta }$

${\rm cosec\; }\theta =\frac{1}{\sin \theta }$

$\cos \theta =\frac{1}{\sec \theta }$

$\sec \theta =\frac{1}{\cos \theta }$

$\tan \theta =\frac{1}{\cot \theta } =\frac{\sin \theta }{\cos \theta }$

$\cot \theta =\frac{1}{\tan \theta } =\frac{\cos \theta }{\sin \theta }$

Type – II

$1=\sin ^{2} \theta +\cos ^{2} \theta$

$1-\sin ^{2} \theta =\cos ^{2} \theta$

$1-\cos ^{2} \theta =\sin ^{2} \theta$

${\rm cosec}^{2} \theta =1+\cot ^{2} \theta$

${\rm cosec}^{2} \theta -\cot ^{2} \theta =1$

${\rm cosec}^{2} \theta -1=\cot ^{2} \theta$

$\sec ^{2} \theta =1+\tan ^{2} \theta$

$\sec ^{2} \theta -\tan ^{2} \theta =1$

$\sec ^{2} \theta -1=\tan ^{2} \theta$

Type – III

$\sin (A+B)=\sin A\cos B+\cos A\sin B$

$\cos (A+B)=\cos A\cos B-\sin A\sin B$

$\cos (A-B)=\cos A\cos B+\sin A\sin B$

$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$

$\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$

$\cot (A+B)=\frac{\cot A\cot B-1}{\cot B+\cot A}$

$\cot (A-B)=\frac{\cot A\cot B+1}{\cot B-\cot A}$

Type – IV

$\sin (A+B)+\sin (A-B)=2\sin A\cos B$

$\sin (A+B)-\sin (A-B)=2\cos A\sin B$

$\cos (A+B)+\cos (A-B)=2\cos A\cos B$

$\cos (A+B)-\cos (A-B)=-2\sin A\sin B$

Type – V

$\sin 3\, \theta \, =3\sin \theta -4\sin ^{3} \theta$

$\cos 3\, \theta =4\cos ^{3} \theta -3\cos \theta$

$\tan 3\, \theta =\frac{3\tan \theta -\tan ^{3} \theta }{1-3\tan ^{2} \theta }$

Type – VI

$1-\cos \theta =2\sin ^{2} \frac{\theta }{2}$

$1+\cos \theta =2\cos ^{2} \frac{\theta }{2}$

$\tan \frac{\theta }{2} \, \, =\, \, \sqrt{\frac{1-\cos \theta }{1+\cos \theta } }$

$\frac{1-\tan \theta }{1+\tan \theta } =\tan \left(\frac{\pi }{4} -\theta \right)$

$\frac{1+\tan \theta }{1-\tan \theta } =\tan \left(\frac{\pi }{4} +\theta \right)$

$1 + sin\theta = \left ( \cos \frac{\theta}{2} + \sin \frac{\theta}{2} \right ) ^2$

$1 – sin\theta = \left ( \cos \frac{\theta}{2} – \sin \frac{\theta}{2} \right ) ^2$

Type – VII

$\sin A+\sin B=2\sin \left(\frac{A+B}{2} \right)\, \, \cos \left(\frac{A-B}{2} \right)$

$\sin A-\sin B=2\cos \left(\frac{A+B}{2} \right)\, \, \sin \left(\frac{A-B}{2} \right)$

$\cos A+\cos B=2\cos \left(\frac{A+B}{2} \right)\, \, \cos \left(\frac{A-B}{2} \right)$

$\cos A-\cos B=-2\sin \left(\frac{A+B}{2} \right)\, \, \sin \left(\frac{A-B}{2} \right)$

Type – VIII

$\sin 2\, \theta =2\sin \theta \cos \theta =\frac{2\tan \theta }{1+\tan ^{2} \theta }$

$\cos 2\, \theta =\cos ^{2} \theta -\sin ^{2} \theta \, =2\cos ^{2} \theta -1\, =1-2\sin ^{2} \theta \, =\frac{1-\tan ^{2} \theta }{1+\tan ^{2} \theta }$

$\tan 2\, \theta =\frac{2\tan \theta }{1-\tan ^{2} \theta }$

Type – IX

$\tan ^{-1} A+\tan ^{-1} B=\tan ^{-1} \left(\frac{A+B}{1-AB} \right)$

$\tan ^{-1} A-\tan ^{-1} B=\tan ^{-1} \left(\frac{A-B}{1+AB} \right)$

Type – X

$sin^{-1}\theta + cos^{-1}\theta = \frac{\pi}{2}$

$tan^{-1}\theta + cot^{-1}\theta = \frac{\pi}{2}$

$sec^{-1}\theta + cosec^{-1}\theta = \frac{\pi}{2}$

Type – XI

$sin^{-1}(-\theta)= -sin^{-1 }\theta$

$tan^{-1}(-\theta)= -tan^{-1 }\theta$

$cosec^{-1}(-\theta)= -cosec^{-1 }\theta$

$cos^{-1}(-\theta)= \pi – cos^{-1 }\theta$

$sec^{-1}(-\theta)= \pi – sec^{-1 }\theta$

$cot^{-1}(-\theta)= \pi – cot^{-1 }\theta$

Type – XII

$sin^{-1}\theta = cosec^{-1}{\frac{1}{\theta}}$

$cosec^{-1}\theta = sin^{-1}{\frac{1}{\theta}}$

$cos^{-1}\theta = sec^{-1}{\frac{1}{\theta}}$

$sec^{-1}\theta = cos^{-1}{\frac{1}{\theta}}$

$tan^{-1}\theta = cot^{-1}{\frac{1}{\theta}}$

$cot^{-1}\theta = tan^{-1}{\frac{1}{\theta}}$

Type – XIII

$2\tan ^{-1} \theta =\sin ^{-1} \left(\frac{2\theta }{1+\theta ^{2} } \right)\, \, =\, \, \cos ^{-1} \left(\frac{1-\theta ^{2} }{1+\theta ^{2} } \right)\, \, \, =\, \, \tan ^{-1} \left(\frac{2\theta }{1-\theta ^{2} } \right)$

## PART - 1

Play Video

Summary of this lecture:
Representation of inverse trigonometric functions

Composite functions identities of trigonometry and inverse trigonometric functions

Inverse Trigonometric Functions identities for negative inputs

Relationship between $\sin^{-1}x,$ $\cos^{-1}x,$ $\tan^{-1}x,$ $\sec^{-1}x,$ $\cosec^{-1}x,$ $\cot^{-1}x$

## PART - 2

Play Video

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

## PART - 3

Play Video

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

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)

## PART - 4

Play Video

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)

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