Difficulty: Medium
Correct Answer: tan 2A
Explanation:
Introduction / Context:
This problem tests your ability to manipulate trigonometric expressions by converting cotangent and tangent into sine and cosine. The goal is to recognise a pattern that simplifies to a familiar double angle formula. Such simplifications are very useful in trigonometry, especially for reducing complex expressions in competitive exams.
Given Data / Assumptions:
Concept / Approach:
The main idea is to rewrite cot A and tan A in terms of sine and cosine. By doing that, you can combine them into a single fraction. You will see cos^2 A − sin^2 A in the numerator and sin A cos A in the denominator, which directly connects to the double angle identities for sine and cosine. Then, dividing appropriately, the expression reduces neatly to tan 2A.
Step-by-Step Solution:
Verification / Alternative check:
You can check this result by taking a specific angle value, for example A = 30 degrees. Compute the original expression numerically and compare it with tan 60 degrees. Using approximate values, you will find both evaluations match, confirming that the identity holds for that angle and hence in general.
Why Other Options Are Wrong:
The expression sin A cos A appears in the numerator during simplification but only after being multiplied by 2 and then divided by cos 2A. The expressions sin^2 A and cos^2 A do not appear as final simplified forms. The option cot 2A is the reciprocal of the correct result and would occur only if the fraction were inverted at the last step, which is not the case.
Common Pitfalls:
Students often forget to combine the fractional terms correctly or misapply the double angle identities, sometimes writing sin 2A = sin^2 A + cos^2 A, which is wrong. Another mistake is to invert the final fraction accidentally, leading to cot 2A instead of tan 2A. Careful algebra and a clear memory of the double angle formulas are essential.
Final Answer:
The simplified form of 2 / (cot A − tan A) is tan 2A.
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