Find the value of the following
`log_((sec^2 60^@-tan^2 60^@))cos60^@`

Answer

To solve the problem \( \log_{\left(\sec^2 60^\circ - \tan^2 60^\circ\right)} \cos 60^\circ \), we will follow these steps:

### Step 1: Calculate \( \sec^2 60^\circ \) and \( \tan^2 60^\circ \)

We know the values of \( \sec 60^\circ \) and \( \tan 60^\circ \):
- \( \sec 60^\circ = \frac{1}{\cos 60^\circ} = \frac{1}{\frac{1}{2}} = 2 \)
- \( \tan 60^\circ = \sqrt{3} \)

Now, we can find:
- \( \sec^2 60^\circ = (2)^2 = 4 \)
- \( \tan^2 60^\circ = (\sqrt{3})^2 = 3 \)

### Step 2: Substitute into the expression

Now substitute these values into the expression \( \sec^2 60^\circ - \tan^2 60^\circ \):
\[ \sec^2 60^\circ - \tan^2 60^\circ = 4 - 3 = 1 \]

### Step 3: Calculate \( \cos 60^\circ \)

Next, we calculate \( \cos 60^\circ \):
\[ \cos 60^\circ = \frac{1}{2} \]

### Step 4: Substitute into the logarithm

Now we substitute these values into the logarithm:
\[ \log_{\left(1\right)} \left(\frac{1}{2}\right) \]

### Step 5: Evaluate the logarithm

The logarithm of any number to the base 1 is undefined. Therefore:
\[ \log_{1} \left(\frac{1}{2}\right) = \text{undefined} \]

### Conclusion

Thus, the value of \( \log_{\left(\sec^2 60^\circ - \tan^2 60^\circ\right)} \cos 60^\circ \) is undefined.

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