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Evaluate each of the following:
\( \sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ} \)
Given:
\( \sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ} \)
To do:
We have to evaluate \( \sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ} \).
Solution:
We know that,
$\sin 45^{\circ}=\frac{1}{\sqrt2}$
$\sin 30^{\circ}=\frac{1}{2}$
$\cos 45^{\circ}=\frac{1}{\sqrt2}$
$\cos 30^{\circ}=\frac{\sqrt3}{2}$
$ \sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ}=\frac{1}{\sqrt2}\times\frac{1}{2}+\frac{1}{\sqrt2}\times\frac{\sqrt3}{2}$
$=\frac{1}{2\sqrt2}+\frac{\sqrt3}{2\sqrt2}$
$=\frac{1+\sqrt3}{2\sqrt2}$
Hence, $ \sin 45^{\circ} \sin 30^{\circ}+\cos 45^{\circ} \cos 30^{\circ}=\frac{1+\sqrt3}{2\sqrt2}$.
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