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

Inverse Trigonometric Functions Lecture 2 Part 1
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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

Inverse Trigonometric Functions 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)

PART - 3

Inverse Trigonometric Functions Lecture 2 Part 3
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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} )

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

Inverse Trigonometric Functions 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)

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