Calculate the value of following :-
`{:cos75^(@)""`

Answer

To calculate the value of \( \cos 75^\circ \), we can use the cosine addition formula. Here’s how to do it step-by-step:

### Step 1: Rewrite \( \cos 75^\circ \)
We can express \( 75^\circ \) as the sum of two angles:
\[ 75^\circ = 45^\circ + 30^\circ \]
Thus, we can write:
\[ \cos 75^\circ = \cos(45^\circ + 30^\circ) \]

### Step 2: Apply the Cosine Addition Formula
The cosine addition formula states:
\[ \cos(a + b) = \cos a \cos b - \sin a \sin b \]
Using this formula, we can write:
\[ \cos 75^\circ = \cos 45^\circ \cos 30^\circ - \sin 45^\circ \sin 30^\circ \]

### Step 3: Substitute Known Values
We know the following trigonometric values:
- \( \cos 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \cos 30^\circ = \frac{\sqrt{3}}{2} \)
- \( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
- \( \sin 30^\circ = \frac{1}{2} \)

Substituting these values into the equation gives:
\[ \cos 75^\circ = \left(\frac{1}{\sqrt{2}}\right) \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{1}{\sqrt{2}}\right) \left(\frac{1}{2}\right) \]

### Step 4: Simplify the Expression
Now, we simplify the expression:
\[ \cos 75^\circ = \frac{\sqrt{3}}{2\sqrt{2}} - \frac{1}{2\sqrt{2}} \]
Combining the two fractions:
\[ \cos 75^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

### Final Result
Thus, the value of \( \cos 75^\circ \) is:
\[ \cos 75^\circ = \frac{\sqrt{3} - 1}{2\sqrt{2}} \]

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