Question

If A, B, C, D are the angles of a quadrilateral, then \[ \frac{\tan A + \tan B + \tan C + \tan D}{\cot A + \cot B + \cot C + \cot D} = \]

• A. \(\cot A\) \(\cot B\) \(\cot C\) \(\cot D\)
• B. \(\tan A\) \(\tan B\) \(\tan C\) \(\tan D\)
• C. \(
  \tan A\) \(\tan B\) \(\tan C\) \(\tan D\)
• D. \(
  \cot A\) \(\cot B\) \(\cot C\) \(\cot D\)

Hint:

For angles in a quadrilateral, the sum of angles is \( 360^\circ \). This property can be used to simplify trigonometric expressions involving the angles.

Answer 1

The Correct Option is B

Step 1: Apply the property of angles in a quadrilateral. The sum of interior angles in a quadrilateral is \( 360^\circ \): \[A + B + C + D = 360^\circ\]
Step 2: Express \(\tan\) and \(\cot\) in terms of each other. Recall the identity \(\cot \theta = \frac{1}{\tan \theta}\).
Step 3: Simplify the provided expression. \[\frac{\tan A + \tan B + \tan C + \tan D}{\cot A + \cot B + \cot C + \cot D} = \frac{\tan A + \tan B + \tan C + \tan D}{\frac{1}{\tan A} + \frac{1}{\tan B} + \frac{1}{\tan C} + \frac{1}{\tan D}}\] \[= \frac{\tan A + \tan B + \tan C + \tan D}{\frac{\tan B \tan C \tan D + \tan A \tan C \tan D + \tan A \tan B \tan D + \tan A \tan B \tan C}{\tan A \tan B \tan C \tan D}}\] \[= \frac{(\tan A + \tan B + \tan C + \tan D) \cdot (\tan A \tan B \tan C \tan D)}{\tan B \tan C \tan D + \tan A \tan C \tan D + \tan A \tan B \tan D + \tan A \tan B \tan C}\] \[= \tan A \tan B \tan C \tan D\]
Step 4: State the final result. The expression simplifies to \(\tan A \tan B \tan C \tan D\).