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**Hint:**In order to determine $\cos 67.5^\circ $ using half-angle identity, first we will determine the $\theta $. Then, choose the accurate half-angle identity and resolve the $ \pm $ sign, substitute the values from the trigonometric values from the trigonometric table, then by further evaluation, we will get the required value

**Complete step by step solution:**

Now, we need to determine $\cos 67.5^\circ $ using half-angle identity.

Therefore, the half-angle identity that we can use here is $1 + \cos \theta = 2{\cos ^2}\dfrac{\theta }{2}$

Let $\dfrac{\theta }{2} = 67.5^\circ $

Then, $\theta = 67.5^\circ \times 2$

$\theta = 135^\circ $

Now, substituting the values in the equation, we have,

$1 + \cos \left( {135^\circ } \right) = 2{\cos ^2}\dfrac{{\left( {135^\circ } \right)}}{2}$

Now, $135^\circ = 180^\circ - 45^\circ $

Thus, $2{\cos ^2}67.5^\circ = 1 + \cos \left( {135^\circ - 45^\circ } \right)$

$2{\cos ^2}67.5^\circ = 1 - \cos 45^\circ $

We know that from trigonometric ratios, $\cos 45^\circ = \dfrac{1}{{\sqrt 2 }}$

Therefore, $2{\cos ^2}67.5^\circ = 1 - \dfrac{1}{{\sqrt 2 }}$

$2{\cos ^2}67.5^\circ = \dfrac{{\sqrt 2 - 1}}{{\sqrt 2 }}$

$2{\cos ^2}67.5^\circ = \dfrac{{\sqrt 2 - 1}}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }}$

$2{\cos ^2}67.5^\circ = \dfrac{{2 - \sqrt 2 }}{2}$

${\cos ^2}67.5^\circ = \dfrac{{2 - \sqrt 2 }}{4}$

$\cos 67.5^\circ = \pm \sqrt {\dfrac{{2 - \sqrt 2 }}{4}} $

Hence,$\cos 67.5^\circ = \pm \dfrac{{\sqrt {2 - \sqrt 2 } }}{2}$

**Note:**Special cases of the sum and difference formulas for sine and cosine are known as the double-angle identities and the half-angle identities.

Let us know about the sum and difference formulas for sine and cosine. Three basic trigonometric identities involve the sums of angles. The functions involved in these identities are sine, cosine and tangent. We can use the angle sum identities to determine the function values of any angles. These identities are useful whenever expressions involving trigonometric functions need to be simplified.

The angle sum identities are

$\cos \left( {a + b} \right) = \cos a\cos b - \sin a\sin b$

$\cos \left( {a - b} \right) = \cos a\cos b + \sin a\sin b$

$\sin \left( {a + b} \right) = \cos a\sin b + \sin a\cos b$

$\sin \left( {a - b} \right) = \cos a\sin b - \sin a\cos b$

$\tan \left( {a + b} \right) = \dfrac{{\tan a + \tan b}}{{1 - \tan a\tan b}}$

$\tan \left( {a - b} \right) = \dfrac{{\tan a - \tan b}}{{1 + \tan a\tan b}}$

Trigonometric table involves the relationship with the length and angles of the triangle. It is generally associated with the right-angled triangle, where one of the angles is always $90^\circ $.

The trigonometric double angle formulas give a relationship between the basic trigonometric functions applied to twice an angle in terms of trigonometric functions of the angle itself. The most common double angle formulas are,

$\sin 2\theta = 2\sin \theta \cos \theta $

$\cos 2\theta = {\cos ^2}\theta - {\sin ^2}\theta $

$\cos 2\theta = 1 - 2{\sin ^2}\theta $

$\cos 2\theta = 2{\cos ^2}\theta - 1$ .

Half-angle formulas allow the expression of trigonometric functions of angles equal to $\dfrac{\theta }{2}$ in terms of $\theta $, which can simplify the functions and make it easier to perform more complex calculations. The most commonly used half angle identities are,

$\cos \left( {\dfrac{\theta }{2}} \right) = \pm \sqrt {\dfrac{{1 + \cos \theta }}{2}} $

$\sin \left( {\dfrac{\theta }{2}} \right) = \pm \sqrt {\dfrac{{1 - \cos \theta }}{2}} $ .

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