Question

\( \csc 48^\circ + \csc 96^\circ + \csc 192^\circ + \csc 384^\circ = \)

• A. \(4\sqrt{3}\)
• B. \(-4\sqrt{3}\)
• C. \(0\)
• D. \(1\)

Hint:

For sums of cosecant terms where the angles are in a geometric progression (e.g., \(\theta, 2\theta, 4\theta, \dots\)), the identity \(cosec } \theta = \cot(\theta/2) - \cot(\theta)\) is highly effective. It often leads to a telescoping sum where most terms cancel out, leaving only the first and last terms.

Answer 1

The Correct Option is C

Step 1: Recall the trigonometric identity for \( \csc \theta \).
A useful identity for sums involving cosecant is: \[ \csc \theta = \cot\left(\frac{\theta}{2}\right) - \cot(\theta). \] Step 2: Apply the identity to each term in the given sum.
Let the given sum be \(S\).
For the first term: \[ \csc 48^\circ = \cot\left(\frac{48^\circ}{2}\right) - \cot(48^\circ) = \cot(24^\circ) - \cot(48^\circ). \] For the second term: \[ \csc 96^\circ = \cot\left(\frac{96^\circ}{2}\right) - \cot(96^\circ) = \cot(48^\circ) - \cot(96^\circ). \] For the third term: \[ \csc 192^\circ = \cot\left(\frac{192^\circ}{2}\right) - \cot(192^\circ) = \cot(96^\circ) - \cot(192^\circ). \] For the fourth term: \[ \csc 384^\circ = \cot\left(\frac{384^\circ}{2}\right) - \cot(384^\circ) = \cot(192^\circ) - \cot(384^\circ). \] Step 3: Sum the expanded terms.
Add all the expanded terms: \[ S = \left(\cot(24^\circ) - \cot(48^\circ)\right) + \left(\cot(48^\circ) - \cot(96^\circ)\right) + \left(\cot(96^\circ) - \cot(192^\circ)\right) + \left(\cot(192^\circ) - \cot(384^\circ)\right). \] Notice that this is a telescoping sum, where intermediate terms cancel out: \[ S = \cot(24^\circ) - \cot(384^\circ). \] Step 4: Simplify the remaining terms using the periodicity of cotangent.
The cotangent function has a period of \(180^\circ\). This means \(\cot(\theta + n \cdot 180^\circ) = \cot(\theta)\) for any integer \(n\).
We can simplify \(\cot(384^\circ)\): \[ \cot(384^\circ) = \cot(384^\circ - 2 \times 180^\circ) = \cot(384^\circ - 360^\circ) = \cot(24^\circ). \] Now substitute this back into the expression for \(S\): \[ S = \cot(24^\circ) - \cot(24^\circ) = 0. \]