If a line makes angles , , with co-ordinate axes prove that 2 + 2… - Vidyadip

Question

If a line makes angles $\alpha, \beta, \gamma$ with co-ordinate axes prove that $\cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma+1=0$

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Answer

$\begin{aligned} & =\left(2 \cos ^2 \alpha-1\right)+\left(2 \cos ^2 \beta-1\right)+\left(2 \cos ^2 \gamma-1\right) \\ & =2\left(\cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma\right)-3 \\ & =2(1)-3 \quad\left[\because \cos ^2 \alpha+\cos ^2 \beta+\cos ^2 \gamma=1\right] \\ & =-1 \\ & \therefore \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma=-1 \\ & \therefore \cos 2 \alpha+\cos 2 \beta+\cos 2 \gamma+1=0\end{aligned}$

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