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What is the simplified value of √((1 - cos A) / (1 + cos A)) expressed using standard trigonometric functions?

Difficulty: Medium

Correct Answer: cosec A - cot A

Explanation:


Introduction / Context:
This problem tests your familiarity with half angle and transformation identities involving cosine. The expression √((1 - cos A) / (1 + cos A)) is a well known form that can be rewritten in terms of tan(A/2), and there is also a direct connection to cosec A and cot A. The aim is to rewrite this square root in an equivalent trigonometric form that matches one of the options.


Given Data / Assumptions:

  • Expression: √((1 - cos A) / (1 + cos A)).
  • Half angle identities relate 1 - cos A and 1 + cos A to sin(A/2) and cos(A/2).
  • A is usually considered in a range where all relevant functions are defined and the square root is real.


Concept / Approach:
First use half angle identities to express 1 - cos A and 1 + cos A in terms of sin(A/2) and cos(A/2). The resulting fraction simplifies to tan^2(A/2), whose square root is tan(A/2). Then use another identity that links tan(A/2) to cosec A and cot A. Specifically, tan(A/2) can be written as cosec A - cot A, which leads directly to the required simplified form.


Step-by-Step Solution:

Step 1: Use half angle identities: 1 - cos A = 2 sin^2(A/2) and 1 + cos A = 2 cos^2(A/2). Step 2: Substitute into the expression: (1 - cos A) / (1 + cos A) = [2 sin^2(A/2)] / [2 cos^2(A/2)]. Step 3: Cancel the factor 2 in numerator and denominator: the fraction becomes sin^2(A/2) / cos^2(A/2). Step 4: This simplifies to (sin(A/2) / cos(A/2))^2 = tan^2(A/2). Step 5: Now take the square root: √((1 - cos A)/(1 + cos A)) = √(tan^2(A/2)) = |tan(A/2)|. Step 6: For typical acute angle ranges, tan(A/2) is positive, so √((1 - cos A)/(1 + cos A)) = tan(A/2). Step 7: Use the identity tan(A/2) = cosec A - cot A. Step 8: Therefore √((1 - cos A)/(1 + cos A)) = cosec A - cot A.


Verification / Alternative check:
To verify, choose A = 60 degrees. On the left side, compute √((1 - cos 60)/(1 + cos 60)) = √((1 - 1/2)/(1 + 1/2)) = √((1/2)/(3/2)) = √(1/3) = 1/√3. On the right side, cosec 60 = 2/√3 and cot 60 = 1/√3, so cosec 60 - cot 60 = 2/√3 - 1/√3 = 1/√3. Both sides match exactly.


Why Other Options Are Wrong:

cosec A + cot A: corresponds instead to cot(A/2) in many identities, not tan(A/2). sec A - cot A and sec A + cot A: mix secant and cotangent and do not match known standard forms for √((1 - cos A)/(1 + cos A)). tan(A/2): is equivalent numerically, but the option list specifically points toward an expression using cosec A and cot A. The correct matching option among those involving cosec and cot is cosec A - cot A.


Common Pitfalls:
Students often mix up which combination of cosec and cot corresponds to tan(A/2) and which corresponds to cot(A/2). Another frequent error is forgetting to take the square root correctly after simplifying to tan^2(A/2). Writing each identity step clearly and checking with a specific angle is a reliable way to confirm the result.


Final Answer:
cosec A - cot A

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