Question

If \(5 \sin \theta = 4\) then the value of \(\frac{\csc \theta - \cot \theta}{\csc \theta + \cot \theta}\) is

• A. -1/4
• B. -1/2
• C. 1/2
• D. 1/4

Hint:

Simplifying the trigonometric expression before substituting values can often make the calculation easier. Here, converting the expression to \((\csc \theta - \cot \theta)^2\) simplifies the problem. Also, recognizing Pythagorean triples (like 3-4-5) can help you quickly find the values of other trig ratios. If \(\sin \theta = 4/5\) (Opp/Hyp), then Adjacent must be 3, so \(\cos \theta = \pm 3/5\) and \(\cot \theta = \pm 3/4\).

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We are given the value of sin θ and asked to find the value of a more complex trigonometric expression involving csc θ and cot θ.

Step 2: Key Formula or Approach (Alternate Method):
Simplify the expression using identities first: (csc θ - cot θ)/(csc θ + cot θ) can be rationalized to (1 - cos θ)/(1 + cos θ). Then substitute cos θ found from sin θ.

Step 3: Detailed Explanation:
Given: 5 sin θ = 4 → sin θ = 4/5. Find cos θ assuming acute angle: cos θ = √(1 - 16/25) = 3/5. Rewrite expression: (csc θ - cot θ)/(csc θ + cot θ) = (1/sin θ - cos θ/sin θ)/(1/sin θ + cos θ/sin θ) = (1 - cos θ)/(1 + cos θ). Substitute cos θ = 3/5: = (1 - 3/5)/(1 + 3/5) = (2/5)/(8/5) = 2/8 = 1/4.

Step 4: Final Answer:
Assuming the standard case of an acute angle, the value of the expression is 1/4.