Curioustab Logo Curioustab
Home » Aptitude » Simplification

If (1 - sin A) / (1 + sin A) = x, then x is equal to which standard trigonometric identity form?

Difficulty: Medium

Correct Answer: (sec A - tan A)^2

Explanation:


Introduction / Context:
This problem tests your understanding of trigonometric identities, especially how to rewrite expressions of the form (1 - sin A) / (1 + sin A) using secant and tangent. Such forms appear in many simplification and transformation questions, and they are directly related to well known identities involving (sec A - tan A) and (sec A + tan A).


Given Data / Assumptions:

  • (1 - sin A) / (1 + sin A) = x.
  • All trigonometric functions are defined for the angle A in the domain considered.
  • We want an equivalent expression involving sec A and tan A.


Concept / Approach:
A classic identity is (sec A - tan A)(sec A + tan A) = 1. From this, one can derive that (sec A - tan A)^2 equals (1 - sin A) / (1 + sin A). The key steps are to convert everything into sine and cosine, simplify, and then recognize the same structure as the left side of the given expression.


Step-by-Step Solution:

Step 1: Recall that sec A = 1 / cos A and tan A = sin A / cos A. Step 2: Consider (sec A - tan A)^2 = (1/cos A - sin A/cos A)^2. Step 3: Combine the terms in the bracket: (1 - sin A) / cos A. Step 4: Square this fraction: (sec A - tan A)^2 = [(1 - sin A)^2] / [cos^2 A]. Step 5: Use the identity cos^2 A = 1 - sin^2 A. Step 6: So (sec A - tan A)^2 = (1 - 2 sin A + sin^2 A) / (1 - sin^2 A). Step 7: Factor the denominator: 1 - sin^2 A = (1 - sin A)(1 + sin A). Step 8: Cancel a common factor of (1 - sin A) from numerator and denominator. Step 9: The expression becomes (1 - sin A) / (1 + sin A). Step 10: Therefore (1 - sin A)/(1 + sin A) = (sec A - tan A)^2, so x = (sec A - tan A)^2.


Verification / Alternative check:
You can choose a specific acute angle, for example A = 30 degrees. Evaluate both the left hand side (1 - sin 30) / (1 + sin 30) and the right hand side (sec 30 - tan 30)^2 numerically. Both computations will give the same value, which confirms the derived identity is correct.


Why Other Options Are Wrong:

sec A - tan A: represents only the square root of the required expression, not the full value. (cosec A - cot A)^2 and cosec A - cot A: are linked to expressions involving cos A rather than sin A, specifically (1 - cos A)/(1 + cos A). (sec A + tan A)^2: is the reciprocal of (sec A - tan A)^2 and simplifies to (1 + sin A)/(1 - sin A), not the given expression.


Common Pitfalls:
Confusing the identities for sine and cosine, or incorrectly squaring and cancelling factors is very common in such questions. It is important to write every step carefully, especially while factoring expressions like 1 - sin^2 A. Mismanaging signs in the expansion of (1 - sin A)^2 is another typical source of error.


Final Answer:
(sec A - tan A)^2

← Previous Question Next Question→

More Questions from Simplification

Discussion & Comments

No comments yet. Be the first to comment!
Join Discussion