Evaluate :
`2 cos ^(2) 157.5^(@) -1`

Answer

To evaluate the expression \( 2 \cos^2(157.5^\circ) - 1 \), we can follow these steps:

### Step 1: Rewrite the angle
We can express \( 157.5^\circ \) in terms of a related angle in the first quadrant. We know that:
\[ 157.5^\circ = 180^\circ - 22.5^\circ \]
Thus, we can rewrite the cosine function:
\[ \cos(157.5^\circ) = \cos(180^\circ - 22.5^\circ) = -\cos(22.5^\circ) \]

### Step 2: Substitute into the expression
Now we substitute this back into our original expression:
\[ 2 \cos^2(157.5^\circ) - 1 = 2 (-\cos(22.5^\circ))^2 - 1 \]
This simplifies to:
\[ 2 \cos^2(22.5^\circ) - 1 \]

### Step 3: Use the double angle formula
We can use the double angle formula for cosine, which states that:
\[ \cos(2\theta) = 2 \cos^2(\theta) - 1 \]
In our case, let \( \theta = 22.5^\circ \):
\[ 2 \cos^2(22.5^\circ) - 1 = \cos(2 \times 22.5^\circ) = \cos(45^\circ) \]

### Step 4: Evaluate \( \cos(45^\circ) \)
We know that:
\[ \cos(45^\circ) = \frac{1}{\sqrt{2}} \]

### Final Answer
Thus, the value of \( 2 \cos^2(157.5^\circ) - 1 \) is:
\[ \frac{1}{\sqrt{2}} \]

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