Question

Assertion :If $\cos \alpha ,\cos \beta$ and $\cos \gamma$ are the direction cosines of any line segment, then ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1$. Reason: If $\cos \alpha ,\cos \beta$ and $\cos \gamma$ are the direction cosines of a line segment, then $\cos 2\alpha +\cos 2\beta +\cos 2\gamma =-1$.

• 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 is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

Obviously, Statement $1$ is true.
$\cos 2\alpha +\cos 2\beta +\cos 2\gamma =2{\cos }^2\alpha -1+2{\cos }^2\beta -1+2{\cos }^2\gamma -1$

$=2({\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma )-3=2-3=-1$

Hence, Statement $2$ is true but does not explain Statement $1$ as it is result derived using the result in the statement.