Question

Assertion :If in a triangle $ABC$, $\cos A\cos B+\sin A\sin B\sin C=1$, then it is an acute angle triangle Reason: In a triangle $ABC$, $\sin A:\sin B:\sin C=a:b:c.$

• 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 D Assertion false but Reason is true

$\cos A\cos B+\sin A\sin B\sin C=1$
$\Rightarrow \sin C=\frac{1-\cos A\cos B}{\sin A\sin B}\le 1(\because -1\le \sin \theta \le 1)$
$\Rightarrow \cos (A-B)\ge 1$
$\Rightarrow \cos (A-B)=1$
$\therefore A=B$
$\therefore \sin C=\frac{1-\cos A\cos B}{\sin A\sin B}=\frac{1-{\cos }^{2}A}{{\sin }^{2}A}=1(\because A=B)$
$\therefore \sin C=1i.e.C=\frac{\pi }{2}$
$\therefore A=B=\frac{\pi }{4}andC=\frac{\pi }{2}$
$\sin A:\sin B:\sin C=\sin \frac{\pi }{4}:\sin \frac{\pi }{4}:\sin \frac{\pi }{2}$
$\sin A:\sin B:\sin C=\frac{1}{\sqrt{2}}:\frac{1}{\sqrt{2}}:1$
$\therefore a:b:c=1:1:\sqrt{2}$
$\Rightarrow$ Assertion (A) is false and Reason is true