Question

Assertion: The value of $\frac{{\tan }^{-1}\frac{4}{3}}{{\tan }^{-1}\frac{1}{2}}$ is equal to $2$.
Reason: $\forall x\in [0,1],{\tan }^{-1}\left(\frac{2x}{1-x^2}\right)=2{\tan }^{-1}x$

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true but R is not the correct explanation of A
• C. R is true but A is false
• D. A is true but R is false

Answer 1

The correct option is A Both A and R are true and R is the correct explanation of A

${\tan }^{-1}\left(\frac{4}{3}\right)$$=2{\tan }^{-1}\left(\frac{1}{2}\right)$ ....Given

${\tan }^{-1}\left(\frac{4}{3}\right)$$={\tan }^{-1}\left(\frac{2\left(\frac{1}{2}\right)}{1-\frac{1}{4}}\right)$
${\tan }^{-1}\left(\frac{4}{3}\right)$$={\tan }^{-1}\frac{4}{3}$

So, LHS = RHS
Hence, the assertion is true.

Now, ${\tan }^{-1}\left(\frac{2x}{1-x^2}\right)$$=2{\tan }^{-1}\left(x\right)$
Hence, the reason is also true.

And R is the correct explanation of the assertion.