If x = tan 2A - tan 2B, then which of the following expressions correctly represents x in terms of sine and cosine of 2A and 2B?

Introduction / Context:
Trigonometry often involves converting between different trigonometric functions to simplify expressions or prove identities. One standard identity expresses the difference of tangents in terms of sine and cosine. In this question, the variable x is defined as tan 2A minus tan 2B, and you are asked to rewrite x using sine and cosine of 2A and 2B. This tests your ability to manipulate trigonometric ratios and apply basic identities correctly.

Given Data / Assumptions:

• x is defined as x = tan 2A - tan 2B.
• Angles 2A and 2B are such that the functions are defined and cos 2A and cos 2B are nonzero.
• We use the basic identity tan theta = sin theta / cos theta.
• Our task is to choose the correct expression for x from the given options.

Concept / Approach:
The key idea is that tan theta is the ratio of sine to cosine. When subtracting two tangent values, you can write each tangent as a sine over cosine and then use the rule for subtracting fractions with different denominators. Specifically, tan 2A - tan 2B becomes a single fraction whose numerator is sin 2A cos 2B minus cos 2A sin 2B and whose denominator is cos 2A cos 2B. Then, by recognising the sine of a difference identity, we can identify the numerator as sin(2A - 2B), giving a compact expression.

Step-by-Step Solution:
Step 1: Start with x = tan 2A - tan 2B. Step 2: Replace each tangent with sine over cosine: tan 2A = sin 2A / cos 2A and tan 2B = sin 2B / cos 2B. Step 3: Write x as a difference of fractions: x = (sin 2A / cos 2A) - (sin 2B / cos 2B). Step 4: Use a common denominator cos 2A cos 2B, so x = (sin 2A cos 2B - cos 2A sin 2B) / (cos 2A cos 2B). Step 5: Recognise the numerator as the identity for sine of a difference: sin(2A - 2B) = sin 2A cos 2B - cos 2A sin 2B. Step 6: Substitute this identity into the fraction, giving x = sin(2A - 2B) / (cos 2A cos 2B).

Verification / Alternative check:
As a numerical check, you can pick specific angles, for example A = 30 degrees and B = 15 degrees, compute x using tan values, and compare with the value obtained from sin(2A - 2B) divided by cos 2A cos 2B. Both calculations will produce the same numerical result (within rounding error), confirming that the formula is consistent. This type of check is useful when you want to be sure that an algebraic identity has been applied correctly.

Why Other Options Are Wrong:

• Option b uses sin 2A minus sin 2B in the numerator instead of the combination sin 2A cos 2B minus cos 2A sin 2B, so it does not match the difference of tangents formula.
• Option c and option d use products of sines divided by either cos 2A minus cos 2B or cos 2A plus cos 2B, which are unrelated to the identity for tan 2A minus tan 2B.
• Option e uses the sum sin 2A plus sin 2B, which corresponds to a different sine identity and not to the difference of tangents.

Common Pitfalls:
Students often confuse the identity for tan A plus tan B or tan A minus tan B with the formulas for sine and cosine of sums and differences. Another common mistake is to combine fractions incorrectly, leading to sign errors or missing factors in the numerator or denominator. To avoid these errors, always write tangent as sine over cosine, carefully use a common denominator, and then match the resulting numerator with the correct sine of a difference or sum identity. Writing each intermediate step clearly helps keep the algebra under control.

Final Answer:
The correct expression for x is sin(2A - 2B) / (cos 2A cos 2B).