Question

Assertion :If $A+B+C=\pi$ then Statement 1: ${\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C$ has its minimum value $\frac{3}{4}$ Reason: Statement 2: Maximum value of $\cos A\cos B\cos C$ is $\frac{1}{8}$

• A. Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1
• B. Both the statements are TRUE and STATEMENT 2 is NOT the correct explanation of STATEMENT 1
• C. STATEMENT 1 is TRUE and STATEMENT 2 is FALSE
• D. STATEMENT 1 is FALSE and STATEMENT 2 is TRUE

Answer 1

The correct option is A Both the statements are TRUE and STATEMENT 2 is the correct explanation of STATEMENT 1

In a triangle, we have $\cos 2A+\cos 2B+\cos 2C=-1-4\cos A\cos B\cos C$
$\Rightarrow {\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C=1-2\cos A\cos B\cos C$
The minimum value of L.H.S. will be achieved when maximum value of R.H.S. will be achieved.
$\cos A\cos B\cos C$ has its value maximum when all three angles are equal and thus the value becomes $\frac{1}{8}.$
Thus, minimum value of L.H.S. is $1-\frac{1}{4}=\frac{3}{4}$