Question

If \( \tan \theta + \sin \theta = a \) and \( \tan \theta - \sin \theta = b \), then the values of \( \cot \theta \) and \( \csc \theta \) are respectively:

• A. \( \frac{1}{a+b}, \frac{1}{a-b} \)
• B. \( \frac{2}{a+b}, \frac{2}{a-b} \)
• C. \( \frac{2}{a-b}, \frac{2}{a+b} \)
• D. \( \frac{1}{a-b}, \frac{1}{a+b} \)

Hint:

When solving trigonometric equations, try adding or subtracting the given equations to eliminate terms and simplify the expressions.

Answer 1

The Correct Option is B

Step 1: Express \( \tan \theta \) and \( \sin \theta \) in terms of \( a \) and \( b \).
We are given two equations: \[ \tan \theta + \sin \theta = a \quad \text{and} \quad \tan \theta - \sin \theta = b \] Adding these two equations: \[ 2 \tan \theta = a + b \quad \Rightarrow \quad \tan \theta = \frac{a + b}{2} \] Subtracting the two equations: \[ 2 \sin \theta = a - b \quad \Rightarrow \quad \sin \theta = \frac{a - b}{2} \] Step 2: Find \( \cot \theta \) and \( \csc \theta \).
We know that \( \cot \theta = \frac{1}{\tan \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). For \( \cot \theta \): \[ \cot \theta = \frac{1}{\tan \theta} = \frac{2}{a + b} \] For \( \csc \theta \): \[ \csc \theta = \frac{1}{\sin \theta} = \frac{2}{a - b} \] Step 3: Conclusion.
Thus, the values of \( \cot \theta \) and \( \csc \theta \) are \( \boxed{\frac{2}{a+b}} \) and \( \boxed{\frac{2}{a-b}} \), respectively.