If a line in space makes angles alpha, beta,a | vedclass

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(C) We know that for a line making angles $\alpha, \beta, \gamma$ with the coordinate axes,the direction cosines satisfy $\cos^2 \alpha + \cos^2 \beta

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$1$

Vedclass · Answer

(C) We know that for a line making angles $\alpha, \beta, \gamma$ with the coordinate axes,the direction cosines satisfy $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$. Given expression: $E = \cos 2\alpha + \cos 2\beta + \cos 2\gamma + \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$. Using the identity $\cos 2	heta = \cos^2 	heta - \sin^2 	heta$,we have: $E = (\cos^2 \alpha - \sin^2 \alpha) + (\cos^2 \beta - \sin^2 \beta) + (\cos^2 \gamma - \sin^2 \gamma) + \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$. Simplifying the expression: $E = \cos^2 \alpha - \sin^2 \alpha + \cos^2 \beta - \sin^2 \beta + \cos^2 \gamma - \sin^2 \gamma + \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$. Canceling the $\sin^2$ terms: $E = \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma$. Since $\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$,the value of the expression is $1$.

If a line in space makes angles $\alpha, \beta$,and $\gamma$ with the coordinate axes,then $\cos 2\alpha + \cos 2\beta + \cos 2\gamma + \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma$ equals:

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