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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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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 $\sin y=\tan ^{2} \frac{x}{2}$