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If tan A = x, then x is equal to which expression written in terms of sec A or cosec A?

Difficulty: Easy

Correct Answer: √(sec^2 A - 1)

Explanation:


Introduction / Context:
This question is designed to test your understanding of fundamental trigonometric identities, especially the Pythagorean identity involving tan A and sec A. When tan A is expressed in terms of sec A, you must recall that sec^2 A and tan^2 A are related in a simple way, which lets you write tan A as a square root expression involving sec^2 A.


Given Data / Assumptions:

  • tan A = x.
  • The identity sec^2 A = 1 + tan^2 A holds for all A where tan and sec are defined.
  • A is typically considered as an acute angle in many aptitude problems, so tan A is a real number.


Concept / Approach:
The key identity is sec^2 A = 1 + tan^2 A. From this, we can isolate tan^2 A in terms of sec^2 A and then take the square root to express tan A as a function of sec A. The square root introduces a plus or minus sign, but in the context of many aptitude questions with standard ranges for A, we usually choose the principal value consistent with the problem setting.


Step-by-Step Solution:

Step 1: Recall the identity sec^2 A = 1 + tan^2 A. Step 2: Rearrange to express tan^2 A in terms of sec^2 A: tan^2 A = sec^2 A - 1. Step 3: Since tan A = x, we have x^2 = tan^2 A = sec^2 A - 1. Step 4: Therefore x = ±√(sec^2 A - 1). Step 5: In the context of the options, the principal square root √(sec^2 A - 1) is taken as the standard expression matching tan A in magnitude. Step 6: So x can be written as √(sec^2 A - 1), which matches one of the provided options exactly.


Verification / Alternative check:
Take a specific angle such as A = 45 degrees. Then tan 45° = 1, so x = 1. Also, sec 45° = √2, so sec^2 45° = 2. Now compute √(sec^2 45° - 1) = √(2 - 1) = √1 = 1, which matches tan 45°. This confirms that the identity and the chosen option are correct.


Why Other Options Are Wrong:

√(cosec^2 A + 1) and √(cosec^2 A - 1): are related to cot A and not to tan A. √(sec^2 A + 1): gives a value larger than sec A and has no direct simple relation to tan A. √(1 - sec^2 A): would involve the square root of a negative number for many angles where sec^2 A is greater than 1, and does not represent tan A.


Common Pitfalls:
Students sometimes confuse the identities sec^2 A = 1 + tan^2 A and cosec^2 A = 1 + cot^2 A, or mix up which identity uses plus or minus. Another common mistake is to assume tan^2 A = 1 - sec^2 A, which is incorrect and reverses the signs. Remembering the correct Pythagorean forms is essential here.


Final Answer:
√(sec^2 A - 1)

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