Question

Assertion (A): If $A+B+C={180}^{\circ }$, then ${\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C=1-2cosA\cos B\cos C$.
Reason (R): If $A+B+C={180}^{\circ }$, then $\cos 2A+\cos 2B+\cos 2C=-1-4\cos A\cos B\cos C$.

• A. A is true, R is true and R is correct explanation of A.
• B. A is true, R is true and R is not a correct explanation of A.
• C. A is true, R is false.
• D. A is false, R is true.

Answer 1

The correct option is A A is true, R is true and R is correct explanation of A.

Let $I=\cos 2A+\cos 2B+\cos 2C$

As $A+B+C={180}^{\circ }$

$\Rightarrow 2C={360}^{\circ }-2(A+B)$, then

$I=2\cos (A+B)\cos (A-B)+\cos ({360}^{\circ }-2(A+B\left)\right)$

$=2\cos (A+B)\cos (A-B)+\cos \left(2\right(A+B\left)\right)$

$=2\cos (A+B)\cos (A-B)+2{\cos }^2(A+B)-1$

$=2\cos (A+B)\left(\cos \right(A-B)+\cos (A+B\left)\right)-1$

Again using $A+B+C={180}^{\circ }\Rightarrow A+B={180}^{\circ }-C$ and the formula of $\cos c+\cos d=2\cos \frac{c+d}{2}\cdot \cos \frac{c-d}{2}$

We get

$I=2\cos ({180}^{\circ }-C)\left(2\cos A\cos B\right)-1.......\left(1\right)$

$=-4\cos A\cos B\cos C-1$

Hence, reason is correct.

Assertion:

As $\cos 2x=2{\cos }^{2}x-1\Rightarrow {\cos }^{2}x=\frac{\cos 2x+1}{2}$

We get

$J={\cos }^{2}A+{\cos }^{2}B+{\cos }^{2}C$

$=\frac{1}{2}(\cos 2A+1+\cos 2B+1+\cos 2C+1)$

$=\frac{1}{2}(\cos 2A+\cos 2B+\cos 2C+3)$

$J=\frac{1}{2}(-4\cos A\cos B\cos C-1+3)$ (using $\left(1\right)$)

$J=1-2\cos A\cos B\cos C$

Hence, assertion is also correct and reason is the correct explanation for assertion.