We know that if \(\alpha\), \(\beta\), and \(\gamma\) are the angles a line makes with the x, y, and z axes respectively, then:
\[cos^2\alpha + cos^2\beta + cos^2\gamma = 1\]This means option (A) is true.
Now, let's analyze option (B):
\[sin^2\alpha + sin^2\beta + sin^2\gamma = (1 - cos^2\alpha) + (1 - cos^2\beta) + (1 - cos^2\gamma)\] \[= 3 - (cos^2\alpha + cos^2\beta + cos^2\gamma) = 3 - 1 = 2\]So, option (B) is also true.
Next, let's analyze option (C):
\[cos~2\alpha + cos~2\beta + cos~2\gamma = (2cos^2\alpha - 1) + (2cos^2\beta - 1) + (2cos^2\gamma - 1)\] \[= 2(cos^2\alpha + cos^2\beta + cos^2\gamma) - 3 = 2(1) - 3 = -1\]So, option (C) is also true.
Finally, let's consider option (D):
There is no direct relationship that states \(cos~\alpha + cos~\beta + cos~\gamma = 1\). This statement is generally false.
Correct Answer: \(cos~\alpha+cos~\beta+cos~\gamma=1\)
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