Question

Assertion :${\tan }^{-1}\frac{2}{5}+{\tan }^{-1}\frac{3}{7}=\frac{\pi }{4}$ Reason: For $x,y,>0{\tan }^{-1}\left(\frac{x}{y}\right)+{\tan }^{-1}\left(\frac{y-x}{y+x}\right)=\frac{\pi }{4}$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion false but Reason is true

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

Consider, ${\tan }^{-1}\left(\frac{2}{5}\right)+{\tan }^{-1}\left(\frac{3}{7}\right)$
$={\tan }^{-1}\left(\frac{\frac{2}{5}+\frac{3}{7}}{1-\frac{6}{35}}\right)$
$={\tan }^{-1}\left(\frac{29}{29}\right)$
$\Rightarrow {\tan }^{-1}\left(\frac{2}{5}\right)+{\tan }^{-1}\left(\frac{3}{7}\right)={\tan }^{-1}1=\frac{\pi }{4}$
And for $x=2y=5,\frac{y-x}{y+x}=\frac{5-2}{5+2}=\frac{3}{7}$
${\tan }^{-1}\left(\frac{x}{y}\right)+{\tan }^{-1}\left(\frac{y-x}{y+x}\right)=\frac{\pi }{4}$
$\therefore$ Reason (R) is also correct with correct explanation of Assertion (A).