If `tan 62^@ = P/Q`, then `tan 28^@` is equal to

Options

A. `P/Q`
B. `Q/P`
C. `(P^2 -Q^2)/P`
D. `Q/P^2`

Answer

To solve the problem, we need to find the value of \( \tan 28^\circ \) given that \( \tan 62^\circ = \frac{P}{Q} \).

### Step-by-step Solution:

1. **Understanding the Relationship**:
We know that \( 62^\circ + 28^\circ = 90^\circ \). This means that \( 28^\circ \) is the complementary angle of \( 62^\circ \).

2. **Using the Complementary Angle Identity**:
The tangent of an angle and its complementary angle are related by the identity:
\[ \tan(90^\circ - \theta) = \cot(\theta) \]
Therefore, we can express \( \tan 28^\circ \) as:
\[ \tan 28^\circ = \tan(90^\circ - 62^\circ) = \cot 62^\circ \]

3. **Expressing Cotangent in Terms of Tangent**:
The cotangent of an angle is the reciprocal of the tangent of that angle:
\[ \cot \theta = \frac{1}{\tan \theta} \]
Thus, we have:
\[ \cot 62^\circ = \frac{1}{\tan 62^\circ} \]

4. **Substituting the Given Value**:
Since we know that \( \tan 62^\circ = \frac{P}{Q} \), we can substitute this into our expression for \( \cot 62^\circ \):
\[ \tan 28^\circ = \cot 62^\circ = \frac{1}{\tan 62^\circ} = \frac{1}{\frac{P}{Q}} = \frac{Q}{P} \]

5. **Final Answer**:
Therefore, we conclude that:
\[ \tan 28^\circ = \frac{Q}{P} \]

### Summary:
If \( \tan 62^\circ = \frac{P}{Q} \), then \( \tan 28^\circ = \frac{Q}{P} \).