Question

The value of $\tan\left[\sin^{-1}\left(\frac{5}{13}\right) + \cot^{-1}\left(\frac{4}{3}\right)\right]$ is

• A. $\frac{26}{11}$
• B. $\frac{56}{33}$
• C. $\frac{63}{41}$
• D. $\frac{65}{43}$
• E. $\frac{32}{13}$

Hint:

Convert inverse trig values into triangles first, then apply identities.

Answer 1

The Correct Option is D

Step 1: Let angles.
\[ \theta = \sin^{-1}\left(\frac{5}{13}\right) \Rightarrow \sin\theta = \frac{5}{13} \] Construct triangle: \[ \text{Opposite} = 5, \text{Hypotenuse} = 13 \] \[ \text{Adjacent} = 12 \] \[ \tan\theta = \frac{5}{12} \]

Step 2: Second angle.
\[ \phi = \cot^{-1}\left(\frac{4}{3}\right) \Rightarrow \cot\phi = \frac{4}{3} \Rightarrow \tan\phi = \frac{3}{4} \]

Step 3: Use tangent addition formula.
\[ \tan(\theta+\phi) = \frac{\tan\theta + \tan\phi}{1 - \tan\theta \tan\phi} \] \[ = \frac{\frac{5}{12} + \frac{3}{4}}{1 - \frac{5}{12}\cdot \frac{3}{4}} \]

Step 4: Simplify numerator.
\[ \frac{5}{12} + \frac{9}{12} = \frac{14}{12} = \frac{7}{6} \]

Step 5: Simplify denominator.
\[ 1 - \frac{15}{48} = \frac{33}{48} \]

Step 6: Final value.
\[ \tan(\theta+\phi) = \frac{7}{6} \div \frac{33}{48} = \frac{7}{6} \cdot \frac{48}{33} = \frac{56}{33} \] \[ \boxed{\frac{56}{33}} \]