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Study the Assertion and Reason and choose the correct option.
If tan⁻¹ x + tan⁻¹ y = π/4, then x + y + xy = 1. Which one of the following is correct?
Assertion: If $\tan^{-1} x + \tan^{-1} y = \frac{\pi}{4}$, then $x + y + xy = 1$.
Reason: $\tan^{-1} x + \tan^{-1} y = \tan^{-1}\left(\frac{x+y}{1-xy}\right)$, $xy < 1$.
Which one of the following is correct?
- aBoth Assertion and Reason are true and Reason is the correct explanation for Assertion.
- bBoth Assertion and Reason are true but Reason is not the correct explanation for Assertion.
- cAssertion is true and Reason is false.
- dAssertion is false and Reason is true.
Answer
Answer
Correct option: a
Answer: (a) Both Assertion and Reason are true and Reason is the correct explanation for Assertion.
By the Reason, $\tan^{-1}x+\tan^{-1}y=\tan^{-1}\frac{x+y}{1-xy}=\frac{\pi}{4}$ gives $\frac{x+y}{1-xy}=1$, so $x+y=1-xy$, i.e. $x+y+xy=1$.