Step-by-step explanation:
[tex]cos \: \theta = \frac{4}{5} ..(given) \\ \\ \sin \: \theta = \sqrt{1 - {cos}^{2} \theta } \\ = \sqrt{1 - \bigg( \frac{4}{5} \bigg)^{2} } \\ = \sqrt{1 - \frac{16}{25} } \\ = \sqrt{ \frac{25 - 16}{25} } \\ = \sqrt{ \frac{9}{25} } \\ \therefore \sin \: \theta = \frac{3}{5} \\ \\ \sin \: 2\theta = 2\sin \: \theta \times \cos \: \theta \\ = 2 \times \frac{3}{5} \times \frac{4}{5} \\ \\ \huge \purple{ \boxed{ \therefore \sin \: 2\theta = \frac{24}{25} }}[/tex]
