What is the value of `(cos10^(@)-sin10^(@))/(cos10^(@)+sin10^(@))`?

Options

A. `tan35^(@)`
B. `tan10^(@)`
C. `(1)/(sqrt(2))`
D. 1

Explanation

To find the value of the expression \((\cos 10^\circ - \sin 10^\circ) / (\cos 10^\circ + \sin 10^\circ)\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{\cos 10^\circ - \sin 10^\circ}{\cos 10^\circ + \sin 10^\circ} \]

### Step 2: Divide numerator and denominator by \(\cos 10^\circ\)
To simplify the expression, we can divide both the numerator and the denominator by \(\cos 10^\circ\):
\[ \frac{1 - \tan 10^\circ}{1 + \tan 10^\circ} \]

### Step 3: Use the tangent subtraction formula
We can recognize that this expression resembles the tangent subtraction formula:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Here, we can let \(A = 45^\circ\) and \(B = 10^\circ\). Since \(\tan 45^\circ = 1\), we can rewrite our expression as:
\[ \tan(45^\circ - 10^\circ) = \tan 35^\circ \]

### Step 4: Conclusion
Thus, we find that:
\[ \frac{\cos 10^\circ - \sin 10^\circ}{\cos 10^\circ + \sin 10^\circ} = \tan 35^\circ \]

### Final Answer
The value of the expression is \(\tan 35^\circ\).

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