Question
Evaluate:
\[ \tan\left(\cos^{-1}\left(\frac{1}{5\sqrt{2}}\right) – \sin^{-1}\left(\frac{4}{\sqrt{17}}\right)\right) \]
Solution
Let
\[ A = \cos^{-1}\left(\frac{1}{5\sqrt{2}}\right), \quad B = \sin^{-1}\left(\frac{4}{\sqrt{17}}\right) \]
Use identity:
\[ \tan(A – B) = \frac{\tan A – \tan B}{1 + \tan A \tan B} \]
Find tan A:
\[ \cos A = \frac{1}{5\sqrt{2}} \Rightarrow \sin A = \sqrt{1 – \frac{1}{50}} = \frac{7}{5\sqrt{2}} \]
\[ \tan A = \frac{\sin A}{\cos A} = \frac{7}{1} = 7 \]
Find tan B:
\[ \sin B = \frac{4}{\sqrt{17}} \Rightarrow \cos B = \sqrt{1 – \frac{16}{17}} = \frac{1}{\sqrt{17}} \]
\[ \tan B = \frac{4}{1} = 4 \]
Now apply formula:
\[ \tan(A – B) = \frac{7 – 4}{1 + 7 \cdot 4} \]
\[ = \frac{3}{29} \]
Final Answer:
\[ \boxed{\frac{3}{29}} \]
Key Concept
Convert inverse trig into triangle values and apply tangent subtraction identity.