Question : In $\triangle{XYZ}$, right-angled at $Y$, if $\sin X = \frac{1}{2}$, find the value of $\cos X \cos Z + \sin X \sin Z$.
Option 1: $\frac{\sqrt{3}}{2}$
Option 2: $\frac{\sqrt{3}}{4}$
Option 3: $\frac{2}{\sqrt{3}}$
Option 4: $\sqrt{3}$
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Correct Answer: $\frac{\sqrt{3}}{2}$
Solution :
Given: In $\triangle {XYZ}$, right-angled at $Y$.
We know that the sum of all the angles in a triangle is 180°.
$\sin X = \frac{1}{2}$
⇒ $\sin X = \sin{30°}$
So, $\angle X = 30°$
Then $\angle Z = 60°$, because $\angle Y + \angle X + \angle Z = 180°$
Then, $\cos X \cos Z + \sin X \sin Z$
$=\cos 30° \cos 60° + \sin 30° \sin 60°$
$=\frac{\sqrt{3}}{2} × \frac{1}{2} + \frac{1}{2} × \frac{\sqrt{3}}{2}$
$ = \frac{\sqrt{3}}{2}$
Hence, the correct answer is $\frac{\sqrt{3}}{2}$.
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