Question

If \(\text{cosec } t + \cot t = \frac{5}{2}\), then the value of \(\tan t\) is equal to

• A. \(\frac{10}{21}\)
• B. \(\frac{21}{20}\)
• C. \(\frac{20}{21}\)
• D. \(\frac{21}{10}\)
• E. \(\frac{10}{42}\)

Hint:

The pairs \((\text{cosec } \theta + \cot \theta)\) and \((\text{cosec } \theta - \cot \theta)\) are reciprocals of each other. The same property holds for \((\sec \theta + \tan \theta)\) and \((\sec \theta - \tan \theta)\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This problem uses the fundamental trigonometric identity relating cosecant and cotangent.

Step 2: Key Formula or Approach:
We know that \(\text{cosec}^2 t - \cot^2 t = 1\).
This can be factored as \((\text{cosec } t + \cot t)(\text{cosec } t - \cot t) = 1\).

Step 3: Detailed Explanation:
Given:
\[ \text{cosec } t + \cot t = \frac{5}{2} \dots (i) \]
Since their product is 1:
\[ \text{cosec } t - \cot t = \frac{1}{5/2} = \frac{2}{5} \dots (ii) \]
Subtract equation (ii) from equation (i) to find \(\cot t\):
\[ (\text{cosec } t + \cot t) - (\text{cosec } t - \cot t) = \frac{5}{2} - \frac{2}{5} \]
\[ 2 \cot t = \frac{25 - 4}{10} = \frac{21}{10} \]
\[ \cot t = \frac{21}{20} \]
Since \(\tan t = \frac{1}{\cot t}\):
\[ \tan t = \frac{20}{21} \]

Step 4: Final Answer:
The value of \(\tan t\) is \(\frac{20}{21}\).