A line makes angles $\frac{\alpha}{2}, \frac{\beta}{2}, \frac{Y}{2}$ with positive directions of the co-ordinate axes $(\mathrm{x}, \mathrm{y}, \mathrm{z}$ axes respectively), then $\cos \alpha+\cos \beta+\cos \gamma$ has the value

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Aicharya · Accepted Answer

$\begin{aligned} & \cos \alpha+\cos \beta+\cos \gamma=2 \cos ^2 \frac{\alpha}{2}-1+2 \cos ^2 \frac{\beta}{2}-1+2 \cos ^2 \frac{\gamma}{2}-1 \ & =2\left(\cos ^2 \frac{\alpha}{2}+\cos ^2 \frac{\beta}{2}+\cos ^2 \frac{\gamma}{2}ight)-3 \ & =2 	imes 1-3\left[\because \cos ^2 \frac{\alpha}{2}+\cos ^2 \frac{\beta}{2}+\cos ^2 \frac{\gamma}{2}=1ight] \ & =-1\end{aligned}$

A line makes angles $\frac{\alpha}{2}, \frac{\beta}{2}, \frac{Y}{2}$ with positive directions of the…

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