Question

If a line makes angles \( \alpha, \beta, \gamma \) with the x-axis, y-axis, and z-axis respectively,
then prove that: \[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = 2 \]

Answer 1

Given: A line makes angles \( \alpha, \beta, \gamma \) with the x-, y-, and z-axes respectively.
We know:
The direction cosines identity:
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \]
Now, use the identity: \[ \sin^2 \theta = 1 - \cos^2 \theta \] So, \[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = (1 - \cos^2 \alpha) + (1 - \cos^2 \beta) + (1 - \cos^2 \gamma) \] \[ = 3 - (\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma) = 3 - 1 = 2 \]
Hence proved:
\[ \sin^2 \alpha + \sin^2 \beta + \sin^2 \gamma = \boxed{2} \]