Question

The value of \[ \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{2}{3} \right) + \cot^{-1} \left( \frac{9}{7} \right) \] is equal to

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{3} \)
• D. \( \frac{\pi}{2} \)
• E. \( 0 \)

Hint:

Use the sum identity for arctangents and the relationship between arctangent and arccotangent to simplify expressions efficiently.

Answer 1

The Correct Option is D

Step 1: Use the Identity for Sum of Two Arctangents Using the identity: \[ \tan^{-1} a + \tan^{-1} b = \tan^{-1} \left( \frac{a + b}{1 - ab} \right) \quad {if} \quad ab<1 \] Setting \( a = \frac{1}{3} \) and \( b = \frac{2}{3} \): \[ \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{2}{3} \right) = \tan^{-1} \left( \frac{\frac{1}{3} + \frac{2}{3}}{1 - \frac{1}{3} \times \frac{2}{3}} \right) \] \[ = \tan^{-1} \left( \frac{1}{1 - \frac{2}{9}} \right) = \tan^{-1} \left( \frac{1}{\frac{7}{9}} \right) = \tan^{-1} \left( \frac{9}{7} \right) \] Step 2: Use the Cotangent-Arctangent Identity Using the identity: \[ \tan^{-1} x + \cot^{-1} x = \frac{\pi}{2} \] for \( x = \frac{9}{7} \), we get: \[ \tan^{-1} \left( \frac{9}{7} \right) + \cot^{-1} \left( \frac{9}{7} \right) = \frac{\pi}{2} \] Since we already computed: \[ \tan^{-1} \left( \frac{1}{3} \right) + \tan^{-1} \left( \frac{2}{3} \right) = \tan^{-1} \left( \frac{9}{7} \right) \] Adding \( \cot^{-1} \left( \frac{9}{7} \right) \): \[ \tan^{-1} \left( \frac{9}{7} \right) + \cot^{-1} \left( \frac{9}{7} \right) = \frac{\pi}{2} \] 
Final Answer: \[ \boxed{\frac{\pi}{2}} \]