Question

Evaluate ${\tan }^{-1}5+{\tan }^{-1}3-{\cot }^{-1}\left(\frac{4}{7}\right)$:

• A. $\frac{\mathrm{\pi }}{3}$
• B. $\frac{\mathrm{\pi }}{4}$
• C. $\frac{\mathrm{\pi }}{2}$
• D. $\frac{\mathrm{\pi }}{6}$

Answer 1

The correct option is C

$\frac{\mathrm{\pi }}{2}$

Explanation for correct option

Evaluating the given trigonometric expression:

Given,

${\tan }^{-1}5+{\tan }^{-1}3-{\cot }^{-1}\left(\frac{4}{7}\right)$

Let, $\mathrm{y}={\tan }^{-1}5+{\tan }^{-1}3-{\cot }^{-1}\left(\frac{4}{7}\right).........\left(1\right)$

We know that: ${\tan }^{-1}\mathrm{a}+{\tan }^{-1}\mathrm{b}={\tan }^{-1}\left[\frac{\mathrm{a}+\mathrm{b}}{1-\mathrm{a}\times \mathrm{b}}\right]$

Applying the above formula in equation $\left(1\right)$:

$\Rightarrow \mathrm{y}=\mathrm{\pi }+{\tan }^{-1}\left[\frac{5+3}{1-5\times 3}\right]-{\cot }^{-1}\left(\frac{4}{7}\right)$

$=\mathrm{\pi }+{\tan }^{-1}\left[\frac{8}{-14}\right]-{\cot }^{-1}\left(\frac{4}{7}\right)$

$=\mathrm{\pi }+{\tan }^{-1}\left(\frac{-4}{7}\right)-{\cot }^{-1}\left(\frac{4}{7}\right)$

$=\mathrm{\pi }-{\tan }^{-1}\left(\frac{4}{7}\right)-{\cot }^{-1}\left(\frac{4}{7}\right)\left[\because -{\tan }^{-1}\theta ={\tan }^{-1}(-\theta )\right]$

$=\mathrm{\pi }-\left[{\tan }^{-1}\left(\frac{4}{7}\right)+{\cot }^{-1}\left(\frac{4}{7}\right)\right]$

$=\mathrm{\pi }-\frac{\mathrm{\pi }}{2}\left(\because {\tan }^{-1}\theta +{\cot }^{-1}\theta =\frac{\pi }{2}\right)$

$=\frac{\mathrm{\pi }}{2}$

So, the value is $\frac{\mathrm{\pi }}{2}$.

Hence, the correct answer is option (C).