Evaluate each of the following:$ \cos 60^{\circ} \cos 45^{\circ}-\sin 60^{\circ} \sin 45^{\circ} $

Given:

\( \cos 60^{\circ} \cos 45^{\circ}-\sin 60^{\circ} \sin 45^{\circ} \)

To do:

We have to evaluate \( \cos 60^{\circ} \cos 45^{\circ}-\sin 60^{\circ} \sin 45^{\circ} \).

Solution:  

We know that,

$\sin 45^{\circ}=\frac{1}{\sqrt2}$

$\sin 60^{\circ}=\frac{\sqrt3}{2}$

$\cos 45^{\circ}=\frac{1}{\sqrt2}$

$\cos 60^{\circ}=\frac{1}{2}$

$ \cos 60^{\circ} \cos 45^{\circ}-\sin 60^{\circ} \sin 45^{\circ}=\frac{1}{2}\times\frac{1}{\sqrt2}-\frac{\sqrt3}{2}\times\frac{1}{\sqrt2}$

$=\frac{1}{2\sqrt2}-\frac{\sqrt3}{2\sqrt2}$

$=\frac{1-\sqrt3}{2\sqrt2}$

Hence, $ \cos 60^{\circ} \cos 45^{\circ}-\sin 60^{\circ} \sin 45^{\circ}=\frac{1-\sqrt3}{2\sqrt2}$.

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Updated on: 10-Oct-2022

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