A line makes angles α, β, γ with the coordinate axes. If α + β =...

We have, $\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma = 1$

$⇒ \cos^{2}\alpha + \cos^{2}\left(\dfrac{\pi}{2} - \alpha\right) + \cos^{2}\gamma = 1$ [given $\alpha + \beta = \dfrac{\pi}{2}$]

$⇒ \cos^{2}\alpha + \sin^{2}\alpha + \cos^{2}\gamma = 1$

$⇒ 1 + \cos^{2}\gamma = 1$

$⇒ \cos^{2}\gamma = 0$

$⇒ cos\gamma = 0$

$∴ \left(cos\alpha + cos\beta + cos\gamma\right)^{2} = \left(cos\alpha + sin\alpha\right)^{2}$

$= 1 + 2sin\alpha \cdot cos\alpha = 1 + \sin 2\alpha$