Question

If  \[\sin \alpha = \frac{4}{5} \text{ and }  \cos \beta = \frac{5}{13}\] , prove that \[\cos\frac{\alpha - \beta}{2} = \frac{8}{\sqrt{65}}\]

 

Answer 1

Given:

\[\sin \alpha = \frac{4}{5} \]

\[\cos \beta = \frac{5}{13}\] .

Now,

\[\cos\alpha = \sqrt{1 - \sin^2 \alpha} = \sqrt{1 - \left( \frac{4}{5} \right)^2} = \frac{3}{5}\]
\[\text{ And } , \]
\[sin\beta = \sqrt{1 - \cos^2 \alpha} = \sqrt{1 - \left( \frac{5}{13} \right)^2} = \frac{12}{13}\]

Now,

\[\cos\left( \alpha - \beta \right) = cos\alpha \times cos\beta + sin\alpha \times sin\beta\]
\[ \Rightarrow \cos\left( \alpha - \beta \right) = \frac{3}{5} \times \frac{5}{13} \times \frac{4}{5} \times \frac{12}{13} = \frac{63}{65}\]
\[\text{ Thus } , \]
\[\cos\frac{\alpha - \beta}{2} = \sqrt{\frac{1 + \cos\left( \alpha - \beta \right)}{2}}\]
\[ = \sqrt{\frac{1 + \frac{63}{65}}{2}}\]
\[ = \frac{8}{\sqrt{65}}\]