Trigonometry & Inverse Trigonometry Question - None

Question

If $\cot\alpha = 1$ and $\cot\alpha$, $\cot(\alpha - \beta)$ and $\cot\beta$ are in A.P., then $\tan\beta$ equals:

Accepted Answer

Step 1: Since cot α, cot(α-β), cot β are in A.P., we have:

2·cot(α-β) = cot α + cot β

Step 2: Use the cotangent subtraction formula: cot(α-β) = (cot α · cot β + 1)/(cot β - cot α)

Step 3: Substitute into the A.P. condition:

2·(cot α · cot β + 1)/(cot β - cot α) = cot α + cot β

Step 4: Substitute cot α = 1:

2·(cot β + 1)/(cot β - 1) = 1 + cot β

Step 5: Cross-multiply:

2(cot β + 1) = (1 + cot β)(cot β - 1)

2 cot β + 2 = cot²β - 1

cot²β - 2 cot β - 3 = 0

Step 6: Factor: (cot β - 3)(cot β + 1) = 0

So cot β = 3 or cot β = -1

Step 7: Therefore: tan β = 1/cot β = 1/3 or tan β = -1

∴ Answer: tan β = 1/3 (or -1 depending on the options provided)

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