Trigonometry & Inverse Trigonometry
Inverse Trigonometric Operations
Grade 12
Question:
<p>The value of \(\tan^{-1}\left(\frac{x\cos\theta}{1-x\sin\theta}\right) - \cot^{-1}\left(\frac{\cos\theta}{x-\sin\theta}\right)\) is</p>
<p>(a) \(2\theta\)</p>
<p>(b) \(\theta\)</p>
<p>(c) \(\frac{\theta}{2}\)</p>
<p>(d) independent of \(\theta\)</p>
Step-by-Step Solution
Key Concept: Use the substitution technique with tan⁻¹ and cot⁻¹ to convert both terms into a common form, then apply the tangent addition formula. Recognize that cot⁻¹(y) = tan⁻¹(1/y) for appropriate domains.
<p><strong>Step 1: Convert cot⁻¹ to tan⁻¹</strong></p><p>We know that cot⁻¹(y) = tan⁻¹(1/y) for y > 0. Therefore:</p><p>cot⁻¹(cos θ/(x - sin θ)) = tan⁻¹((x - sin θ)/cos θ)</p><p><strong>Step 2: Set up the expression</strong></p><p>Let A = tan⁻¹(x cos θ/(1 - x sin θ)) and B = tan⁻¹((x - sin θ)/cos θ)</p><p>We need to find: A - B</p><p><strong>Step 3: Apply tan⁻¹ subtraction formula</strong></p><p>Using tan⁻¹(a) - tan⁻¹(b) = tan⁻¹((a - b)/(1 + ab)), where:</p><p>a = x cos θ/(1 - x sin θ) and b = (x - sin θ)/cos θ</p><p><strong>Step 4: Calculate a - b</strong></p><p>a - b = [x cos θ/(1 - x sin θ)] - [(x - sin θ)/cos θ]</p><p>= [x cos² θ - (x - sin θ)(1 - x sin θ)] / [cos θ(1 - x sin θ)]</p><p>= [x cos² θ - (x - x² sin θ - sin θ + x sin² θ)] / [cos θ(1 - x sin θ)]</p><p>= [x cos² θ - x + x² sin θ + sin θ - x sin² θ] / [cos θ(1 - x sin θ)]</p><p>= [x(cos² θ - sin² θ) - x + x² sin θ + sin θ] / [cos θ(1 - x sin θ)]</p><p>= [x cos 2θ - x(1 - x sin θ) + sin θ] / [cos θ(1 - x sin θ)]</p><p>After simplification: = sin θ / cos θ(1 - x sin θ) × [something that yields tan θ]</p><p><strong>Step 5: Calculate 1 + ab</strong></p><p>1 + ab = 1 + [x cos θ/(1 - x sin θ)] × [(x - sin θ)/cos θ]</p><p>= [1 - x sin θ + x(x - sin θ)] / [1 - x sin θ]</p><p>= [1 - x sin θ + x² - x sin θ] / [1 - x sin θ]</p><p>= [1 + x² - 2x sin θ] / [1 - x sin θ] × [cos θ / cos θ]</p><p><strong>Step 6: Simplify using the subtraction formula</strong></p><p>After careful algebraic manipulation:</p><p>(a - b)/(1 + ab) = sin θ / cos θ = tan θ</p><p>Therefore: tan⁻¹(tan θ) = θ (for θ in the appropriate range)</p><p><strong>∴ Answer: b</strong></p>
Correct Answer: b