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If the angles made by a straight line with the coordinate axes are $\alpha, \frac{\pi}{2}-\alpha, \beta$, then $\beta$ is equal to
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Verified Answer
The correct answer is:
$\frac{\pi}{2}$
We know that, if $\alpha, \beta, \gamma$ are angles of a line which makes from the coordinate axes
But, given $\alpha=\alpha, \beta=\frac{\pi}{2}-\alpha, \gamma=\beta$
From Eq. (i)
$\cos ^2 \alpha+\cos ^2\left(\frac{\pi}{2}-\alpha\right)+\cos ^2 \beta=1$
$\begin{array}{cc}\Rightarrow & \left(\cos ^2 \alpha+\sin ^2 \alpha\right)+\cos ^2 \beta=1 \\ \Rightarrow & \cos ^2 \beta=0 \\ \Rightarrow & \cos \beta=0=\cos \frac{\pi}{2} \\ \Rightarrow & \beta=\frac{\pi}{2}\end{array}$
But, given $\alpha=\alpha, \beta=\frac{\pi}{2}-\alpha, \gamma=\beta$
From Eq. (i)
$\cos ^2 \alpha+\cos ^2\left(\frac{\pi}{2}-\alpha\right)+\cos ^2 \beta=1$
$\begin{array}{cc}\Rightarrow & \left(\cos ^2 \alpha+\sin ^2 \alpha\right)+\cos ^2 \beta=1 \\ \Rightarrow & \cos ^2 \beta=0 \\ \Rightarrow & \cos \beta=0=\cos \frac{\pi}{2} \\ \Rightarrow & \beta=\frac{\pi}{2}\end{array}$
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