A line makes the same angle with each of the x and z axes.

Question

A line makes the same angle $	heta$ with each of the $x$ and $z$ axes. If the angle $\beta$, which it makes with the $y$-axis, is such that $\sin^2\beta=3\sin^2	heta$, then $\cos^2	heta$ equals

Accepted Answer

Direction cosines $(\cos\theta,\cos\beta,\cos\theta)$: $2\cos^2\theta+\cos^2\beta=1\Rightarrow 2\cos^2\theta=\sin^2\beta=3\sin^2\theta=3(1-\cos^2\theta)$.
$$2\cos^2\theta=3-3\cos^2\theta\implies5\cos^2\theta=3\implies\cos^2\theta=\frac35.$$

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