Trigonometry & Inverse Trigonometry
Inverse Trigonometric Functions
Grade 12
Question:
<p>If \(a_1, a_2, a_3, \ldots, a_n\) are in AP with common difference \(5\) and if \(a_i a_j \neq -1\) for \(i, j = 1, 2, \ldots, n\), then \(\tan^{-1}\left(\frac{5}{1 + a_1 a_2}\right) + \tan^{-1}\left(\frac{5}{1 + a_2 a_3}\right) + \tan^{-1}\left(\frac{5}{1 + a_{n-1}a_n}\right) + \ldots + \tan^{-1}\left(\frac{5}{1 + a_n a_1}\right)\) is equal to</p>
<p>(a) \(\tan^{-1}\left(\frac{c_2 - c_1}{1 + c_2 c_1}\right)\)</p>
<p>(b) \(\tan^{-1}\left(\frac{c_3 - c_2}{1 + c_3 c_2}\right)\)</p>
<p>(c) \(\tan^{-1}\left(\frac{c_n}{1 + a_n a_{n-1}}\right)\)</p>
<p>(d) \(\tan^{-1}\left(\frac{5a_1}{1 + a_n a_{n-1}}\right)\)</p>
Step-by-Step Solution
Key Concept: Use the telescoping property of inverse tangent: tan⁻¹(x) - tan⁻¹(y) = tan⁻¹((x-y)/(1+xy)). Recognize that each term tan⁻¹(5/(1+aᵢaᵢ₊₁)) can be written as tan⁻¹(aᵢ₊₁) - tan⁻¹(aᵢ) when the common difference is 5.
<p><strong>Step 1:</strong> Since a₁, a₂, ..., aₙ are in AP with common difference 5, we have: aᵢ₊₁ - aᵢ = 5 for all i.</p><p><strong>Step 2:</strong> Recall the inverse tangent subtraction formula: tan⁻¹(x) - tan⁻¹(y) = tan⁻¹((x-y)/(1+xy)), provided xy > -1.</p><p><strong>Step 3:</strong> Observe that: tan⁻¹(aᵢ₊₁) - tan⁻¹(aᵢ) = tan⁻¹((aᵢ₊₁ - aᵢ)/(1 + aᵢ₊₁·aᵢ)) = tan⁻¹(5/(1 + aᵢaᵢ₊₁))</p><p><strong>Step 4:</strong> Therefore, each term in the given sum equals a difference of consecutive inverse tangent terms: tan⁻¹(5/(1+aᵢaᵢ₊₁)) = tan⁻¹(aᵢ₊₁) - tan⁻¹(aᵢ)</p><p><strong>Step 5:</strong> The sum becomes telescoping: [tan⁻¹(a₂) - tan⁻¹(a₁)] + [tan⁻¹(a₃) - tan⁻¹(a₂)] + ... + [tan⁻¹(aₙ) - tan⁻¹(aₙ₋₁)]</p><p><strong>Step 6:</strong> All intermediate terms cancel, leaving: tan⁻¹(aₙ) - tan⁻¹(a₁) = tan⁻¹((aₙ - a₁)/(1 + aₙa₁))</p><p><strong>Step 7:</strong> Since a₁, a₂, ..., aₙ are in AP with common difference 5: aₙ = a₁ + (n-1)×5, so aₙ - a₁ = (n-1)×5. However, examining the options, they use c₁, c₂, etc., which likely represent these terms. The answer matches option A: tan⁻¹((c₂ - c₁)/(1 + c₂c₁)), where c₁ = a₁ and c₂ = aₙ represent the first and last terms of the sequence.</p><p><strong>∴ Answer:</strong> A</p>
Correct Answer: A