Question
Evaluate:
\[ \sin\left(2\tan^{-1}(0.75)\right) \]
Solution
Let
\[ \theta = \tan^{-1}(0.75) = \tan^{-1}\left(\frac{3}{4}\right) \]
Then,
\[ \tan\theta = \frac{3}{4} \Rightarrow \text{Opposite} = 3,\ \text{Adjacent} = 4 \Rightarrow \text{Hypotenuse} = 5 \]
\[ \sin\theta = \frac{3}{5}, \quad \cos\theta = \frac{4}{5} \]
Now use identity:
\[ \sin 2\theta = 2\sin\theta \cos\theta \]
\[ = 2 \cdot \frac{3}{5} \cdot \frac{4}{5} = \frac{24}{25} \]
Final Answer:
\[ \boxed{\frac{24}{25}} \]
Key Concept
Convert into triangle form and apply double-angle identity.