Difficulty: Easy
Correct Answer: (1 - √2) / √2
Explanation:
Introduction / Context:
This question tests your knowledge of exact trigonometric values at special angles and manipulation of simple surd expressions. The angle π/4 corresponds to 45 degrees, for which the values of sine, cosine and tangent are well known. The goal is to compute cos(π/4) - tan(π/4) exactly and then match this value with one of the given algebraic surd expressions.
Given Data / Assumptions:
Concept / Approach:
First evaluate cos(π/4) and tan(π/4) numerically in terms of square roots. Then subtract to find x. After that, simplify x algebraically and compare it with the provided options. In many aptitude questions, equivalent surd forms are used as answer choices, so this comparison step is important.
Step-by-Step Solution:
Verification / Alternative check:
Approximate the values numerically. cos(π/4) is about 0.707 and tan(π/4) is exactly 1, so x is approximately -0.293. Now compute (1 - √2) / √2 numerically: √2 is about 1.414, so 1 - √2 is about -0.414, and dividing by 1.414 gives roughly -0.293. The two values match closely, confirming the equality.
Why Other Options Are Wrong:
Common Pitfalls:
Students sometimes confuse radians and degrees, but here π/4 is clearly a standard radian angle. Another mistake is to treat 1 as 2/2 instead of matching the surd denominator. Working carefully with surds and keeping the same denominator is crucial for correctly comparing expressions.
Final Answer:
(1 - √2) / √2
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