Trigonometry & Inverse Trigonometry
Trigonometry
star_batch_jee_advanced_2025
Grade 12
Question:
If $2\tan^{-1}\frac{1}{5} - \sin^{-1}\frac{1}{5} = -\cos^{-1}\frac{63}{\lambda}$, then $\lambda =$
Step-by-Step Solution
Key Concept: Use the tangent subtraction formula $\tan^{-1}a - \tan^{-1}b = \tan^{-1}\left(\frac{a-b}{1+ab}\right)$ to combine inverse trigonometric functions.
Starting with $2\tan^{-1}\left(\frac{1}{5}\right) - \sin^{-1}\left(\frac{3}{5}\right)$, rewrite as $\tan^{-1}\left(\frac{5}{12}\right) - \sin^{-1}\left(\frac{3}{5}\right)$. Converting to tangent form: $\tan^{-1}\left(\frac{5}{12}\right) - \tan^{-1}\left(\frac{3}{4}\right) = \tan^{-1}\left(\frac{16}{63}\right) = -\cos^{-1}\left(\frac{63}{65}\right)$, giving $\lambda = 65$.
Correct Answer: 65