Question

Assertion (A):
${\cos }^2\theta +{\cos }^2({60}^0-\theta )+{\cos }^2({60}^0+\theta )=\frac{3}{2}$ .
Reason (R):
$\sin \alpha -\sin ({120}^0-\alpha )+\sin ({120}^0+\alpha )=0$.
Note: $x$ in $\cos x$ is represented in degrees.

• A. The assertion is true, Reason is true and Reason is the correct explanation of Assertion.
• B. The assertion is true, Reason is true and Reason is not the correct explanation of Assertion.
• C. Assertion is true, Reason is false.
• D. Assertion is false, Reason is true.

Answer 1

Answer: (B)

The assertion is true, Reason is true and Reason is not the correct explanation of Assertion.

Assertion (A) :

${\cos }^2\theta +{\cos }^2(60-\theta )+{\cos }^2(60+\theta )$

$={\cos }^2\theta +\left[\right(\cos (60+\theta )+\cos (60-\theta ){)}^2-2\cos (60-\theta )\cos (60+\theta )]$

$={\cos }^2\theta +(2\cos 60\cos \theta {)}^2-(\cos \left(120\right)+\cos 2\theta )$

$={\cos }^2\theta +{\cos }^2\theta -(\cos 2\theta -\frac{1}{2})$

$=2{\cos }^2\theta -\cos 2\theta +\frac{1}{2}$

$=2{\cos }^2\theta -(2{\cos }^2\theta -1)+\frac{1}{2}$

$=\frac{3}{2}$; True

Reason (R) :

$\sin \alpha -\sin (120-\alpha )+\sin (120+\alpha )$

$=\sin \alpha +2\cos \left(120\right)\sin \alpha$

$=\sin \alpha -\sin \alpha =0$; True

As we see, Assertion (A) it cannot be explained by reason (R).

So, A is true, R is true and R is not correct explanation of A.