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\}

Answer:

\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\}

Answer:

\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\}

Answer:

\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} 

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