Question

Evaluate the trigonometric expression: $\tan \left(\cos^{-1}\left(\frac{4}{5}\right) + \tan^{-1}\left(\frac{2}{3}\right)\right) =$

• A. $\frac{17}{6}$
• B. $\frac{17}{3}$
• C. $\frac{18}{5}$
• D. $\frac{7}{15}$

Hint:

Remember the standard $3-4-5$ right triangle! A cosine value of $\frac{4}{5}$ instantly means the corresponding tangent value is $\frac{3}{4}$. Once you have both fractions ($\frac{3}{4}$ and $\frac{2}{3}$), simple arithmetic gives $\frac{0.75 + 0.66}{1 - 0.5} = \frac{1.416}{0.5} \approx 2.833$, which matches $\frac{17}{6}$ perfectly.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question requires us to compute the value of a composite trigonometric function containing different inverse trigonometric functions inside a tangent wrapper.

Step 2: Key Formula or Approach:
1. Convert the inverse cosine function into an inverse tangent function using a standard right-angled triangle definition: $$\cos^{-1}\left(\frac{\text{base}}{\text{hypotenuse}}\right) = \tan^{-1}\left(\frac{\text{perpendicular}}{\text{base}}\right)$$ 2. Apply the tangent addition identity: $$\tan(\tan^{-1} X + \tan^{-1} Y) = \frac{X + Y}{1 - XY}$$

Step 3: Detailed Explanation:
3. Let $\theta = \cos^{-1}\left(\frac{4}{5}\right) \implies \cos\theta = \frac{4}{5}$.
Using the Pythagorean identity, the perpendicular side is $\sqrt{5^2 - 4^2} = 3$. Thus, $\tan\theta = \frac{3}{4} \implies \theta = \tan^{-1}\left(\frac{3}{4}\right)$. 4. Substitute this back into the original expression: $$\tan \left(\tan^{-1}\left(\frac{3}{4}\right) + \tan^{-1}\left(\frac{2}{3}\right)\right)$$ 5. Apply the compound angle formulation rules for tangent: $$= \frac{\frac{3}{4} + \frac{2}{3}}{1 - \left(\frac{3}{4} \times \frac{2}{3}\right)}$$ Calculate the numerator and denominator fractions: $$\text{Numerator} = \frac{9 + 8}{12} = \frac{17}{12}$$ $$\text{Denominator} = 1 - \frac{6}{12} = \frac{6}{12}$$ $$= \frac{\frac{17}{12}}{\frac{6}{12}} = \frac{17}{6}$$

Step 4: Final Answer:
The final evaluated value is $\frac{17}{6}$, which corresponds to option (A).