The expression `(tan 57^@ + cot37^@)/(tan 33^@ + cot 53^@)` is equal to

Options

A. `tan 33^@ cot 57^@`
B. `tan 57^@ cot 37^@`
C. `tan 33^@ cot 53^@`
D. `tan 53^@ cot 37^@`

Answer

To solve the expression \((\tan 57^\circ + \cot 37^\circ) / (\tan 33^\circ + \cot 53^\circ)\), we can use some trigonometric identities and properties.

### Step-by-Step Solution:

1. **Use the cotangent identity**:
We know that \(\cot \theta = \tan(90^\circ - \theta)\). Therefore, we can rewrite \(\cot 37^\circ\) and \(\cot 53^\circ\):
\[ \cot 37^\circ = \tan(90^\circ - 37^\circ) = \tan 53^\circ \]
\[ \cot 53^\circ = \tan(90^\circ - 53^\circ) = \tan 37^\circ \]

2. **Substitute these into the expression**:
Now we can rewrite the original expression:
\[ \frac{\tan 57^\circ + \tan 53^\circ}{\tan 33^\circ + \tan 37^\circ} \]

3. **Use the tangent addition formula**:
We can use the tangent addition formula, which states that \(\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\). However, in this case, we can simply evaluate the sum:
\[ \tan 57^\circ + \tan 53^\circ = \tan 57^\circ + \tan(90^\circ - 37^\circ) = \tan 57^\circ + \cot 37^\circ \]
\[ \tan 33^\circ + \tan 37^\circ = \tan 33^\circ + \tan(90^\circ - 53^\circ) = \tan 33^\circ + \cot 53^\circ \]

4. **Evaluate the expression**:
We can simplify the expression further. Notice that \(\tan 57^\circ = \cot 33^\circ\) and \(\tan 37^\circ = \cot 53^\circ\). Therefore, we can rewrite:
\[ \frac{\tan 57^\circ + \tan 53^\circ}{\tan 33^\circ + \tan 37^\circ} = \frac{\cot 33^\circ + \tan 53^\circ}{\tan 33^\circ + \cot 53^\circ} \]

5. **Recognize the symmetry**:
This expression has a symmetry that allows us to conclude that it simplifies to 1, as the numerator and denominator are equal in terms of their trigonometric identities.

6. **Final Result**:
Thus, the value of the original expression is:
\[ 1 \]