`(cos21^(@) -sin21^(@))/(cos21^(@) + sin21^(@))` A)`tan21^(@)`B)`cot66^(@)`C)`tan42^(@)`D)`cot42^(@)`

Options

A. `tan21^(@)`
B. `cot66^(@)`
C. `tan42^(@)`
D. `cot42^(@)`

Answer

To solve the expression \((\cos 21^\circ - \sin 21^\circ) / (\cos 21^\circ + \sin 21^\circ)\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{\cos 21^\circ - \sin 21^\circ}{\cos 21^\circ + \sin 21^\circ} \]

### Step 2: Factor out \(\cos 21^\circ\) from both numerator and denominator
We can factor \(\cos 21^\circ\) out of both the numerator and the denominator:
\[ = \frac{\cos 21^\circ (1 - \tan 21^\circ)}{\cos 21^\circ (1 + \tan 21^\circ)} \]
Here, we have used the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).

### Step 3: Cancel \(\cos 21^\circ\)
Since \(\cos 21^\circ\) is common in both the numerator and denominator, we can cancel it out (assuming \(\cos 21^\circ \neq 0\)):
\[ = \frac{1 - \tan 21^\circ}{1 + \tan 21^\circ} \]

### Step 4: Use the tangent subtraction formula
We can use the tangent subtraction formula:
\[ \tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B} \]
Here, let \(A = 45^\circ\) and \(B = 21^\circ\):
\[ = \tan(45^\circ - 21^\circ) = \tan 24^\circ \]

### Step 5: Relate \(\tan 24^\circ\) to cotangent
Since we need to express this in terms of cotangent, we can use the identity:
\[ \tan(90^\circ - \theta) = \cot \theta \]
Thus,
\[ \tan 24^\circ = \cot(90^\circ - 24^\circ) = \cot 66^\circ \]

### Final Answer
Therefore, the value of the expression \((\cos 21^\circ - \sin 21^\circ)/(\cos 21^\circ + \sin 21^\circ)\) is:
\[ \cot 66^\circ \]