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

Questions discussed in this lecture

Simplify:
1. $\tan ^{-1} \left(\sqrt{\frac{1-\cos x}{1+\cos x} } \right),\, \, x<\pi$ (NCERT Exercise 2.2 Q7)
2. $\tan ^{-1} \left(\frac{\cos x-\sin x}{\cos x+\sin x} \right),\, x<\pi$ (NCERT Exercise 2.2 Q8)
3. $\cot ^{-1} \left(\frac{1}{\sqrt{x^{2} -1} } \right),\, \, |x|\, >1$ (NCERT Example 6)
4. $\tan ^{-1} \frac{1}{\sqrt{x^{2} -1} } ,\, |x|\, >1$ (NCERT Exercise 2.2 Q6)
5. $\tan ^{-1} \left(\frac{\sqrt{1+x^{2} } -1}{x} \right),\, \, x\ne 0$ (NCERT Exercise 2.2 Q5)

## PART - 2

Play Video

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

## PART - 3

Play Video

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)

वो खुद ही तय करतें हैं मंज़िल आसमानों की,
परिन्दों को नहीं दी जाती तालीम उड़ानों की,
रखते हैं जो हौंसला आसमां छूने का,
उनको नहीं परवाह कभी गिर जाने की |