Question 1: Evaluate:

$\displaystyle \text{(i) } \cos \Bigg\{ \sin^{-1} \Bigg( - \frac{7}{25} \Bigg) \Bigg\}$

$\displaystyle \text{(ii) } \sec \Bigg\{ \cot^{-1} \Bigg( - \frac{5}{12} \Bigg) \Bigg\}$

$\displaystyle \text{(iii) } \cot \Bigg\{ \sec^{-1} \Bigg( - \frac{13}{5} \Bigg) \Bigg\}$

$\displaystyle \text{(i) } \cos \Bigg\{ \sin^{-1} \Bigg( - \frac{7}{25} \Bigg) \Bigg\}$

$\displaystyle = \cos \Bigg\{ -\sin^{-1} \Bigg(\frac{7}{25} \Bigg) \Bigg\}$

$\displaystyle = \cos \Bigg\{ \sin^{-1} \Bigg( \frac{7}{25} \Bigg) \Bigg\}$

$\displaystyle =\cos \Bigg\{ \cos^{-1} \sqrt{ 1 - \Bigg( \frac{7}{25} \Bigg)^2 } \Bigg\}$

$\displaystyle = \cos \Bigg\{ \cos^{-1} \frac{24}{25} \Bigg\}$

$\displaystyle = \frac{24}{25}$

$\displaystyle \text{(ii) } \sec \Bigg\{ \cot^{-1} \Bigg( - \frac{5}{12} \Bigg) \Bigg\}$

$\displaystyle = \sec \Bigg\{ \pi - \cot^{-1} \Bigg( \frac{5}{12} \Bigg) \Bigg\}$

$\displaystyle = \sec \Bigg\{\cot^{-1} \Bigg( \frac{5}{12} \Bigg) \Bigg\}$

$\displaystyle = \sec \Bigg\{\cos^{-1} \Bigg( \frac{1}{1+ \Big( \frac{5}{12} \Big)^2 } \Bigg) \Bigg\}$

$\displaystyle = \sec \Bigg\{\cos^{-1} \Bigg( \frac{5}{13} \Bigg) \Bigg\}$

$\displaystyle = \sec \Bigg\{\sec^{-1} \Bigg( \frac{13}{5} \Bigg) \Bigg\}$

$\displaystyle = \frac{13}{5}$

$\displaystyle \text{(iii) } \cot \Bigg\{ \sec^{-1} \Bigg( - \frac{13}{5} \Bigg) \Bigg\}$

$\displaystyle = \cot \Bigg\{ \sec^{-1} \Bigg( \pi- \frac{13}{5} \Bigg) \Bigg\}$

$\displaystyle = \cot \Bigg\{ \sec^{-1} \Bigg( - \frac{13}{5} \Bigg) \Bigg\}$

$\displaystyle = \cot \Bigg\{ \tan^{-1} \Bigg( \frac{\sqrt{ 1 - \Big( \frac{13}{5} \Big)^2} }{\frac{5}{13}} \Bigg) \Bigg\}$

$\displaystyle = \cot \Bigg\{\tan^{-1} \Bigg( \frac{12}{5} \Bigg) \Bigg\}$

$\displaystyle = \cot \Bigg\{\cot^{-1} \Bigg( \frac{5}{12} \Bigg) \Bigg\}$

$\displaystyle = \frac{5}{12}$

$\\$

Question 2: Evaluate:

$\displaystyle \text{(i) } \tan \Bigg\{ \cos^{-1} \Bigg( - \frac{7}{25} \Bigg) \Bigg\}$

$\displaystyle \text{(ii) } \mathrm{cosec} \Bigg\{ \cot^{-1} \Bigg( - \frac{12}{5} \Bigg) \Bigg\}$

$\displaystyle \text{(iii) } \cos \Bigg\{ \tan^{-1} \Bigg( - \frac{3}{4} \Bigg) \Bigg\}$

$\displaystyle \text{(i) } \tan \Bigg\{ \cos^{-1} \Bigg( - \frac{7}{25} \Bigg) \Bigg\}$

$\displaystyle = \tan \Bigg\{ \cos^{-1} \Bigg( \pi- \frac{7}{25} \Bigg) \Bigg\}$

$\displaystyle = -\tan \Bigg\{ \cos^{-1} \Bigg( \frac{7}{25} \Bigg) \Bigg\}$

$\displaystyle = -\tan \Bigg\{ \tan^{-1} \Bigg( \frac{\sqrt{ 1 - \Big( \frac{7}{25} \Big)^2} }{\frac{7}{25}} \Bigg) \Bigg\}$

$\displaystyle = -\tan \Bigg\{\tan^{-1} \Bigg( \frac{24}{7} \Bigg) \Bigg\}$

$\displaystyle = -\frac{24}{7}$

$\displaystyle \text{(ii) } \mathrm{cosec} \Bigg\{ \cot^{-1} \Bigg( - \frac{12}{5} \Bigg) \Bigg\}$

$\displaystyle = \mathrm{cosec} \Bigg\{ \cot^{-1} \Bigg( \pi- \frac{12}{5} \Bigg) \Bigg\}$

$\displaystyle = \mathrm{cosec} \Bigg\{ \cot^{-1} \Bigg( \frac{12}{5} \Bigg) \Bigg\}$

$\displaystyle = \mathrm{cosec} \Bigg\{ \sin^{-1} \Bigg( \frac{\frac{5}{12}}{\sqrt{ 1 + \Big( \frac{5}{12} \Big)^2} } \Bigg) \Bigg\}$

$\displaystyle = \mathrm{cosec} \Bigg\{ \sin^{-1} \Bigg( \frac{5}{13} \Bigg) \Bigg\}$

$\displaystyle = \mathrm{cosec} \Bigg\{\mathrm{cosec}^{-1} \Bigg( \frac{13}{5} \Bigg) \Bigg\}$

$\displaystyle = \frac{13}{5}$

$\displaystyle \text{(iii) } \cos \Bigg\{ \tan^{-1} \Bigg( - \frac{3}{4} \Bigg) \Bigg\}$

$\displaystyle = \cos \Bigg\{ - \tan^{-1} \Bigg( \frac{3}{4} \Bigg) \Bigg\}$

$\displaystyle = \cos \Bigg\{ \tan^{-1} \Bigg( \frac{3}{4} \Bigg) \Bigg\}$

$\displaystyle = \cos \Bigg( \cos^{-1} \frac{1}{\sqrt{1 + \Big( \frac{3}{4} \Big)^2}} \Bigg)$

$\displaystyle = \cos \Bigg( \cos^{-1} \frac{1}{\sqrt{1 + \frac{9}{16} }} \Bigg)$

$\displaystyle = \cos \Bigg( \cos^{-1} \frac{1}{\sqrt{\frac{25}{16} } } \Bigg)$

$\displaystyle = \cos \Bigg( \cos^{-1} \frac{1}{\frac{5}{4}} \Bigg)$

$\displaystyle = \frac{4}{5}$

$\\$

$\displaystyle \text{Question 3: Evaluate: } \sin \Bigg\{ \cos^{-1} \Bigg( - \frac{3}{5} \Bigg) + \cot^{-1} \Bigg( - \frac{5}{12} \Bigg) \Bigg\}$

$\displaystyle \sin \Bigg\{ \cos^{-1} \Bigg( - \frac{3}{5} \Bigg) + \cot^{-1} \Bigg( - \frac{5}{12} \Bigg) \Bigg\}$

$\displaystyle = \sin \Bigg\{ \pi - \cos^{-1} \Bigg( \frac{3}{5} \Bigg) + \pi - \cot^{-1} \Bigg( \frac{5}{12} \Bigg) \Bigg\}$

$\displaystyle = \sin \Bigg\{ 2\pi - \Bigg[ \cos^{-1} \Bigg( \frac{3}{5} \Bigg) + \cot^{-1} \Bigg( \frac{5}{12} \Bigg) \Bigg] \Bigg\}$

$\displaystyle = - \sin \Bigg\{ \cos^{-1} \Bigg( \frac{3}{5} \Bigg) + \cot^{-1} \Bigg( \frac{5}{12} \Bigg) \Bigg\}$

$\displaystyle = - \sin \Bigg\{ \sin^{-1} \sqrt{1 - \Bigg( \frac{3}{5} \Bigg)^2 } + \sin^{-1} \frac{\frac{12}{5}}{1 + \Bigg( \frac{5}{12} \Bigg)^2} \Bigg\}$

$\displaystyle = - \sin \Bigg\{ \sin^{-1} \frac{4}{5} + \sin^{-1} \frac{12}{13} \Bigg\}$

$\displaystyle = - \Bigg[ \sin \Bigg( \sin^{-1} \frac{4}{5} \Bigg) \cos \Bigg( \sin^{-1} \frac{12}{13} \Bigg) + \cos \Bigg( \sin^{-1} \frac{4}{5} \Bigg) \sin \Bigg( \sin^{-1} \frac{12}{13} \Bigg) \Bigg]$

$\displaystyle = - \Bigg[ \frac{4}{5} \times \sqrt{1 - \Big( \frac{12}{13}\Big)^2 } + \frac{3}{5} \times \frac{12}{13} \Bigg]$

$\displaystyle = - \Bigg[ \frac{4}{5} \times \frac{5}{13}+ \frac{3}{5} \times \frac{12}{13} \Bigg]$

$\displaystyle = - \Bigg[ \frac{20}{65} + \frac{36}{65} \Bigg]$

$\displaystyle = - \frac{56}{65}$