# Properties of Inverse Trigonometric Functions

• Sin−1x + Cos−1 x= \frac{\pi}{2} ∀ x ∈ [−1, 1]
• Tan−1 x+ Cot−1 x= \frac{\pi}{2} ∀ x ∈ R
• Sec−1 x+ Cosec−1 x= \frac{\pi}{2} ∀ x ∈ R − (−1, 1)
• sin−1(−x) = −sin−1x ∀ x ∈ [−1, 1]
cosec−1(−x) = −cosec−1x ∀ x ∈ R − (−1, 1)
tan−1(−x) = −tan−1x ∀ x ∈ R
• cos−1(−x) = π − cos−1x ∀ x ∈ [−1, 1]
sec−1(−x) = π − sec−1x ∀ x ∈ R − (−1, 1)
cot−1(−x) = π − cot−1x ∀ x ∈ R
• \sin^{-1}\left(\frac{1}{x}\right)=cosec^{-1}(x) ∀ x ∈ R − (−1, 1)
• \cos^{-1}\left(\frac{1}{x}\right)=sec^{-1}(x) ∀ x ∈ R − (−1, 1)
• \tan^{-1}\left(\frac{1}{x}\right)=\begin{cases}\cot^{-1}x & \forall \ \ x > 0\\-\pi+\cot^{-1}x & \forall \ \ x < 0\end{cases}
• \sin^{-1}(sin \theta)=\begin{cases}2\pi + \theta & {\tt if} & \frac{-5\pi}{2} \leq \theta \leq \frac{-3\pi}{2}\\-\pi - \theta & {\tt if} & \frac{-3\pi}{2} \leq \theta \leq \frac{-\pi}{2}\\ \theta & {\tt if} & \frac{-\pi}{2} \leq \theta \leq \frac{\pi}{2}\\\pi - \theta & {\tt if} & \frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2}\\ \theta-2\pi & {\tt if} & \frac{3\pi}{2} \leq \theta \leq \frac{5\pi}{2}\\ 3\pi - \theta & {\tt if} & \frac{5\pi}{2} \leq \theta \leq \frac{7\pi}{2} \end{cases} and so on
• \cos^{-1}(cos \theta)=\begin{cases}-2\pi - \theta & {\tt if} & -3\pi \leq \theta \leq -2\pi\\2\pi + \theta & {\tt if} & -2\pi \leq \theta \leq -\pi\\ -\theta & {\tt if} & -\pi \leq \theta \leq 0\\ \theta & {\tt if} & 0 \leq \theta \leq \pi\\2\pi-\theta & {\tt if} & \pi \leq \theta \leq 2\pi\\ \theta-2\pi & {\tt if} & 2\pi \leq \theta \leq 3\pi \end{cases} and so on
• \tan^{-1}(tan \theta)=\begin{cases}2\pi + \theta & {\tt if} & \frac{-5\pi}{2} < \theta < \frac{-3\pi}{2}\\\pi + \theta & {\tt if} & \frac{-3\pi}{2} < \theta < \frac{-\pi}{2}\\ \theta & {\tt if} & \frac{-\pi}{2} < \theta < \frac{\pi}{2}\\ \theta-\pi & {\tt if} & \frac{\pi}{2} < \theta < \frac{3\pi}{2}\\ \theta-2 \pi & {\tt if} & \frac{3\pi}{2} < \theta < \frac{5\pi}{2} \end{cases}
• \cot^{-1}(cot \theta)=\begin{cases}2\pi + \theta & {\tt if} & \theta \ \epsilon \ (-2\pi,-\pi) \\\pi + \theta & {\tt if} & \theta \ \epsilon \ (-\pi,0)\\ \theta & {\tt if} & \theta \ \epsilon \ (0, \pi)\\ \theta-\pi & {\tt if} & \theta \ \epsilon \ (\pi,2\pi)\\ \theta-2 \pi & {\tt if} & \theta \ \epsilon \ (2\pi,3\pi) \end{cases}
• \sec^{-1}(\sec \theta)=\begin{cases}2\pi + \theta & {\tt if} & \theta \ \epsilon \ \left[-2\pi,\frac{-3\pi}{2}\right)\cup \left(\frac{-3\pi}{2},-\pi\right] \\-\theta & {\tt if} & \theta \ \epsilon \ \left[-\pi,\frac{-\pi}{2}\right)\cup \left(\frac{-\pi}{2},0\right]\\ \theta & {\tt if} & \theta \ \epsilon \ \left[0,\frac{\pi}{2}\right)\cup \left(\frac{\pi}{2},\pi\right]\\2\pi-\theta & {\tt if} & \theta \ \epsilon \ \left[\pi,\frac{3\pi}{2}\right)\cup \left(\frac{3\pi}{2},2\pi\right]\\ \theta-2 \pi & {\tt if} & \theta \ \epsilon \ \left[2\pi,\frac{5\pi}{2}\right)\cup \left(\frac{5\pi}{2},3\pi\right] \end{cases}
• cosec^{-1}(cosec \theta)=\begin{cases}2\pi + \theta & {\tt if} & \theta \ \epsilon \ \left[\frac{-5\pi}{2},\frac{-3\pi}{2}\right]- \left\{-2\pi\right\} \\-\pi-\theta & {\tt if} & \theta \ \epsilon \ \left[\frac{-3\pi}{2},\frac{-\pi}{2}\right]- \left\{-\pi\right\}\\ \theta & {\tt if} & \theta \ \epsilon \ \left[\frac{-\pi}{2},\frac{\pi}{2}\right]- \left\{0\right\}\\\pi-\theta & {\tt if} & \theta \ \epsilon \ \left[\frac{\pi}{2},\frac{3\pi}{2}\right]- \left\{\pi\right\}\\ \theta-2 \pi & {\tt if} & \theta \ \epsilon \ \left[\frac{3\pi}{2},\frac{5\pi}{2}\right]- \left\{-2\pi\right\} \end{cases}
• \tan^{-1}x+\tan^{-1}y=\begin{cases}\tan^{-1}\left(\frac{x+y}{1-xy}\right), & {\tt if} & xy < 1\\\pi+\tan^{-1}\left(\frac{x+y}{1-xy}\right), & {\tt if} & xy > 1\\ \frac{\pi}{2}, & {\tt if} & xy = 1\end{cases}
• \tan^{-1}x-\tan^{-1}y=\tan^{-1}\left(\frac{x-y}{1+xy}\right)
• \sin^{-1}x+\sin^{-1}y=\sin^{-1}\left[x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}\right]
• \sin^{-1}x-\sin^{-1}y=\sin^{-1}\left[x\sqrt{1-y^{2}}-y\sqrt{1-x^{2}}\right]
• \cos^{-1}x+\cos^{-1}y=\cos^{-1}\left[xy-\sqrt{(1-x^{2})(1-y^{2})}\right]
• \cos^{-1}x-\cos^{-1}y=\cos^{-1}\left[xy+\sqrt{(1-x^{2})(1-y^{2})}\right]
• 2\sin^{-1}x=\begin{cases}\sin^{-1}(2x\sqrt{1-x^{2}}),& {\tt if} & \frac{-1}{\sqrt{2}} \leq x \leq \frac{1}{\sqrt{2}}\\\pi-\sin^{-1}(2x\sqrt{1-x^{2}}),& {\tt if} & \frac{1}{\sqrt{2}} \leq x \leq 1 \\ -\pi-\sin^{-1}(2x\sqrt{1-x^{2}}),& {\tt if} & -1 \leq x \leq \frac{-1}{\sqrt{2}}\end{cases}
• 2\sin^{-1}x = \cos^{-1}(1-2x^{2})
• 2\cos^{-1}x=\begin{cases}\cos^{-1}(2x^{2}-1),& {\tt if} & 0 \leq x \leq 1\\2\pi-\cos^{-1}(2x^{2}-1),& {\tt if} & -1 \leq x \leq 0\end{cases}
• 2\cos^{-1}x = \sin^{-1}(2x\sqrt{1-x^{2}})
• 3\sin^{-1}x=\begin{cases}\sin^{-1}(3x-4x^{3});& {\tt if} & \frac{-1}{2} \leq x \leq \frac{1}{2}\\\pi-\sin^{-1}(3x-4x^{3});& {\tt if} & \frac{1}{2} < x \leq 1 \\ -\pi-\sin^{-1}(3x-4x^{3}),& {\tt if} & -1 \leq x < \frac{-1}{2}\end{cases}
• 3\cos^{-1}x=\begin{cases}\cos^{-1}(4x^{3}-3x);& {\tt if} & \frac{1}{2} \leq x \leq 1\\2\pi-\cos^{-1}(4x^{3}-3x);& {\tt if} & \frac{-1}{2} \leq x \leq \frac{1}{2} \\ 2\pi+\cos^{-1}(4x^{3}-3x),& {\tt if} & -1 \leq x \leq \frac{-1}{2}\end{cases}
• 2\tan^{-1}x=\begin{cases}\sin^{-1}\left(\frac{2x}{1+x^{2}}\right);& {\tt if} & -1 \leq x \leq 1\\\pi-\sin^{-1}\left(\frac{2x}{1+x^{2}}\right);& {\tt if} & x > 1 \\-\pi-\sin^{-1}\left(\frac{2x}{1+x^{2}}\right),& {\tt if} & x < -1\end{cases}
• 2\tan^{-1}x=\begin{cases}\cos^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right),& {\tt if} & 0 \leq x < \infty\\-\cos^{-1}\left(\frac{1-x^{2}}{1+x^{2}}\right),& {\tt if} & -\infty < x \leq 0\end{cases}
• 2\tan^{-1}x=\begin{cases}\tan^{-1}\left(\frac{2x}{1-x^{2}}\right);& {\tt if} & -1 < x < 1\\\pi+\tan^{-1}\left(\frac{2x}{1-x^{2}}\right);& {\tt if} & x > 1 \\-\pi+\tan^{-1}\left(\frac{2x}{1-x^{2}}\right),& {\tt if} & x < -1\end{cases}
• 3\tan^{-1}x=\begin{cases}\tan^{-1}\left(\frac{3x-x^{3}}{1-3x^{2}}\right);& {\tt if} & \frac{-1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}\\\pi+\tan^{-1}\left(\frac{3x-x^{3}}{1-3x^{2}}\right);& {\tt if} & x > \frac{1}{\sqrt{3}} \\-\pi+\tan^{-1}\left(\frac{3x-x^{3}}{1-3x^{2}}\right);& {\tt if} & x < \frac{-1}{\sqrt{3}} \end{cases}

Short cut methods:

• If tan−1 x + tan−1 y + tan−1 z = \frac{\pi}{2} then xy + yz + zx = 1
• If tan−1 x + tan−1 y + tan−1 z = π then x + y + z = xyz
• If tan−1 x + tan−1 y = \frac{\pi}{2}, then xy = 1
• If cot−1 x + cot−1 y = \frac{\pi}{2}, then xy = 1
• If sin−1 x + sin−1 y + sin−1 z = \frac{\pi}{2}, then x2 + y2 + z2 + 2xyz = 1
• If sin−1 x + sin−1 y + sin−1 z = π, then x\sqrt{1-x^{2}}+y\sqrt{1-y^{2}}+z\sqrt{1-z^{2}}=2xyz
• If sin−1 x + sin−1 y + sin−1 z = \frac{3\pi}{2}, then xy + yz + zx = 3
• If cos−1 x + cos−1 y + cos−1 z = π, then x2 + y2 + z2 + 2xyz = 1
• If cos−1 x + cos−1 y + cos−1 z = 3π, then xy + yz + zx = 3
• If sin−1 x + sin−1 y = θ, then cos−1 x + cos−1 y = π − θ
• If cos−1 x + cos−1 y = θ, then sin−1 x + sin−1 y = π − θ
• If \cos^{-1}\left(\frac{x}{a}\right)+\cos^{-1}\left(\frac{y}{b}\right)=\theta, then \frac{x^{2}}{a^{2}}-\frac{2xy}{ab} \cos\theta + \frac{y^{2}}{b^{2}}=\sin^{2}\theta
• \tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\tan^{-1}\left(\frac{x+y+z-xyz}{1-xy-yz-zx}\right)
• \tan^{-1}(x_{1})+\tan^{-1}(x_{2})+\tan^{-1}(x_{3})+---+ \tan^{-1}(x_{n})=\tan^{-1}\left(\frac{S_{1}-S_{3}+S_{5}-S_{7}+....}{1-S_{2}+S_{4}-S_{6}+S_{8}....}\right)

where S1 = ∑ x1

S2 = ∑ x1 x2

S3 = ∑ x1 x2 x3

S4 = ∑ x1 x2 x3 x4

--------------------

Sn = x1 x2 x3 ..... xn

• If \sin^{-1}\left(\frac{x}{a}\right)+\sin^{-1}\left(\frac{y}{b}\right)=\theta, then \frac{x^{2}}{a^{2}}+\frac{2xy}{ab} \cos\theta + \frac{y^{2}}{b^{2}}=\sin^{2}\theta

\tan^{-1}\left(\frac{1}{1+x(x+1)}\right)+\tan^{-1}\left(\frac{1}{1+(x+1)(x+2)}\right)+---+\tan^{-1}\left(\frac{1}{1+(x+n-1)(x+n)}\right)=\tan^{-1}(x+n)-\tan^{-1}x

### Part3: View the Topic in this video From 00:40 To 56:22

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1. (i) sin−1 (sin θ) = θ;   if θ ∈ \left[-\frac{\pi}{2},\frac{\pi}{2}\right]

(ii) cos−1 (cos θ) = θ;   if θ ∈ [0, π]

(iii) tan−1 (tan θ) = θ;   if θ ∈ \left(-\frac{\pi}{2},\frac{\pi}{2}\right)

(iv) cosec−1 (cosec θ) = θ;   if θ ∈ \left[-\frac{\pi}{2},\frac{\pi}{2}\right],\theta\neq 0

(v) sec−1 (sec θ) = θ;   if θ ∈ [0, π], \theta\neq \frac{\pi}{2}

(vi) cot−1 (cot θ) = θ;   if θ ∈ (0, π)

2. (i) sin−1 (−x) = −sin−1 x;  if x ∈ [−1, 1]

(ii) cos−1 (−x) = π − cos−1 x;  if x ∈ [−1, 1]

(iii) tan−1 (−x) = −tan−1 x;  if x ∈ R

(iv) cosec−1 (−x) = π − cosec−1 x;  if x ∈ (−∞, −1] ∪ [1, ∞)

(v) sec−1 (−x) = π − sec−1 x;  if x ∈ (−∞, −1] ∪ [1, ∞)

(vi) cot−1 (−x) = π − cot−1 x;  if x ∈ R

3. (i) sin (sin−1 x) = x;  if x ∈ [−1, 1]

(ii) cos (cos−1 x) = x;  if x ∈ [−1, 1]

(iii) tan (tan−1 x) = x;  if x ∈ R

(iv) cosec (cosec−1 x) = x;  if x ∈ (−∞, −1] ∪ [1, ∞)

(v) sec (sec−1 x) = x;  if x ∈ (−∞, −1] ∪ [1, ∞)

(vi) cot (cot−1 x) = x;  if x ∈ R

4. (i) sin−1 x + cos−1 x = \frac{\pi}{2}; if x ∈ [−1, 1]

(ii) tan−1 x + cot−1 x =\frac{\pi}{2} ; if x ∈ R

(iii) sec−1 x + cosec−1 x =\frac{\pi}{2} ; if x ∈ (−∞, −1] ∪ [1, ∞)

5. \tan^{-1}x+\tan^{-1}y=\tan^{-1}\left(\frac{x+y}{1-xy}\right); If x > 0, y > 0 and xy < 1

6. \tan^{-1}x+\tan^{-1}y=\pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right); If x > 0, y > 0 and xy > 1

7. \tan^{-1}x+\tan^{-1}y=-\pi + \tan^{-1}\left(\frac{x+y}{1-xy}\right); If x < 0, y < 0 and xy > 1

8. \tan^{-1}x-\tan^{-1}y=\tan^{-1}\left(\frac{x-y}{1+xy}\right); If xy > −1

9. \tan^{-1}x-\tan^{-1}y=\pi + \tan^{-1}\left(\frac{x-y}{1+xy}\right); If x > 0, y < 0 and xy < −1

10. \tan^{-1}x-\tan^{-1}y=-\pi + \tan^{-1}\left(\frac{x-y}{1+xy}\right); If x < 0, y > 0 and xy < −1

11. \tan^{-1}x+\tan^{-1}y+\tan^{-1}z=\tan^{-1}\left[\frac{x+y+z-xyz}{1-xy-yz-zx}\right]

12. \cot^{-1}x+\cot^{-1}y=\cot^{-1}\frac{xy-1}{y+x}

13. \cot^{-1}x-\cot^{-1}y=\cot^{-1}\frac{xy+1}{y-x}

14. \sin^{-1}x+\sin^{-1}y=\sin^{-1}\left\{x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}\right\}; If −1 ≤ x, y ≤ 1 and x2 + y2 ≤ 1 or if xy < 0 and x2 + y2 > 1.

15. \sin^{-1}x+\sin^{-1}y=\pi-\sin^{-1}\left\{x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}\right\}; If 0 < x, y ≤ 1 and x2 + y2 > 1.

16. \sin^{-1}x+\sin^{-1}y=-\pi-\sin^{-1}\left\{x\sqrt{1-y^{2}}+y\sqrt{1-x^{2}}\right\}; If −1 ≤ x, y < 0 and x2 + y2 > 1.

17. \sin^{-1}x-\sin^{-1}y=\sin^{-1}\left\{x\sqrt{1-y^{2}}-y\sqrt{1-x^{2}}\right\}; If −1 ≤ x, y ≤ 1 and x2 + y2 ≤ 1 if or xy > 0 and x2 + y2 > 1.

18. \sin^{-1}x-\sin^{-1}y=\pi-\sin^{-1}\left\{x\sqrt{1-y^{2}}-y\sqrt{1-x^{2}}\right\}; If 0 < x ≤ 1, −1 ≤  y ≤ 0 and x2 + y2 > 1.

19. \sin^{-1}x-\sin^{-1}y=-\pi-\sin^{-1}\left\{x\sqrt{1-y^{2}}-y\sqrt{1-x^{2}}\right\}; If −1 ≤ x < 0, 0 < y ≤ 1 and x2 + y2 > 1.

20. \cos^{-1}x+\cos^{-1}y=\cos^{-1}\left\{xy-\sqrt{1-x^{2}}.\sqrt{1-y^{2}}\right\}; If −1 ≤ x, y ≤ 1 and x + y ≥ 0.

21. \cos^{-1}x+\cos^{-1}y=2\pi-\cos^{-1}\left\{xy-\sqrt{1-x^{2}}.\sqrt{1-y^{2}}\right\}; If −1 ≤ x, y ≤ 1 and x + y ≤ 0.

22. \cos^{-1}x-\cos^{-1}y=\cos^{-1}\left\{xy+\sqrt{1-x^{2}}.\sqrt{1-y^{2}}\right\}; If −1 ≤ x, y ≤ 1 and x ≤ y.

23. \cos^{-1}x-\cos^{-1}y=-\cos^{-1}\left\{xy+\sqrt{1-x^{2}}.\sqrt{1-y^{2}}\right\}; If −1 ≤ y ≤ 0, 0 < x ≤ 1 and x ≥ y.

24. 2\sin^{-1}x=\sin^{-1}\left(2x\sqrt{1-x^{2}}\right), \ {\tt{If}} \ -\frac{1}{\sqrt{2}}\leq x \leq \frac{1}{\sqrt{2}}

25. 2\sin^{-1}x=\pi-\sin^{-1}\left(2x\sqrt{1-x^{2}}\right), \ {\tt{If}} \ \frac{1}{\sqrt{2}}\leq x \leq 1

26. 2\sin^{-1}x=-\pi-\sin^{-1}\left(2x\sqrt{1-x^{2}}\right), \ {\tt{If}} \ -1\leq x \leq \frac{-1}{\sqrt{2}}

27. 3\sin^{-1}x=\sin^{-1}\left(3x-4x^{3}\right), \ {\tt{If}} \ \frac{-1}{2}\leq x \leq \frac{1}{2}

28. 3\sin^{-1}x=\pi-\sin^{-1}\left(3x-4x^{3}\right), \ {\tt{If}} \ \frac{1}{2} < x \leq 1

29. 3\sin^{-1}x=-\pi-\sin^{-1}\left(3x-4x^{3}\right), \ {\tt{If}} \ -1 \leq x < -\frac{1}{2}

30. 2\cos^{-1}x=\cos^{-1}\left(2x^{2}-1\right), \ {\tt{If}} \ 0 \leq x \leq 1

31. 2\cos^{-1}x=2\pi-\cos^{-1}\left(2x^{2}-1\right), \ {\tt{If}} \ -1 \leq x \leq 0

32. 3\cos^{-1}x=\cos^{-1}\left(4x^{3}-3x\right), \ {\tt{If}} \ \frac{1}{2} \leq x \leq 1

33. 3\cos^{-1}x=2\pi-\cos^{-1}\left(4x^{3}-3x\right), \ {\tt{If}} \ -\frac{1}{2} \leq x \leq \frac{1}{2}

34. 3\cos^{-1}x=2\pi+\cos^{-1}\left(4x^{3}-3x\right), \ {\tt{If}} \ -1 \leq x \leq -\frac{1}{2}

35. 2\tan^{-1}x=\tan^{-1}\left(\frac{2x}{1-x^{2}}\right), \ {\tt{If}} \ -1 < x \leq 1

36. 2\tan^{-1}x=\pi+\tan^{-1}\left(\frac{2x}{1-x^{2}}\right), \ {\tt{If}} \ x > 1

37. 2\tan^{-1}x=-\pi+\tan^{-1}\left(\frac{2x}{1-x^{2}}\right), \ {\tt{If}} \ x < -1

38. 2\tan^{-1}x=\sin^{-1}\left(\frac{2x}{1+x^{2}}\right), \ {\tt{If}} \ -1 \leq x \leq 1

39. 2\tan^{-1}x= \pi - \sin^{-1}\left(\frac{2x}{1+x^{2}}\right), \ {\tt{If}} \ x \gt 1

40. 2\tan^{-1}x=-\pi-\sin^{-1}\left(\frac{2x}{1+x^{2}}\right), \ {\tt{If}} \ x < -1

41. 2\tan^{-1}x=\cos^{-1}\left(\frac{1-x^2}{1+x^{2}}\right), \ {\tt{If}} \ 0 \leq x < \infty

42. 2\tan^{-1}x= - \cos^{-1}\left(\frac{1-x^2}{1+x^{2}}\right), \ {\tt{If}} \ -\infty < x \leq 0

43. 3\tan^{-1}x=\tan^{-1}\left(\frac{3x-x^3}{1-3x^{2}}\right), \ {\tt{If}} \ \frac{-1}{\sqrt{3}} < x < \frac{1}{\sqrt{3}}

44. 3\tan^{-1}x=\pi+\tan^{-1}\left(\frac{3x-x^3}{1-3x^{2}}\right), \ {\tt{If}} \ x > \frac{1}{\sqrt{3}}

45. 3\tan^{-1}x=-\pi+\tan^{-1}\left(\frac{3x-x^3}{1-3x^{2}}\right), \ {\tt{If}} \ x < -\frac{1}{\sqrt{3}}