In trigonometric simplification, evaluate the expression [1 - tan(90° - θ) + sec(90° - θ)] / [tan(90° - θ) + sec(90° - θ) + 1]. Express the result as a single trigonometric function of θ.

Introduction / Context:
This question involves cofunction identities and half angle ideas in trigonometry. The expression contains tan(90° - θ) and sec(90° - θ) inside a rational expression. You must simplify it to a compact expression in terms of θ alone. Problems like this test how well you remember the relationships between trigonometric functions of complementary angles and whether you can manipulate a fraction involving several terms in the numerator and denominator.

Given Data / Assumptions:

• Expression: E = [1 - tan(90° - θ) + sec(90° - θ)] / [tan(90° - θ) + sec(90° - θ) + 1].
• All angles are measured in degrees.
• θ is such that all expressions are defined (no division by zero).
• Standard cofunction identities tan(90° - θ) = cot θ and sec(90° - θ) = cosec θ are used.

Concept / Approach:
We first convert the functions of (90° - θ) into functions of θ using cofunction identities: tan(90° - θ) = cot θ, sec(90° - θ) = cosec θ. After this substitution, the expression becomes a rational function in cot θ and cosec θ. We then simplify the numerator and denominator to see if the fraction can be expressed in terms of a half angle function such as tan(θ / 2). In many standard problems of this kind, the final result reduces to a half angle tangent formula.

Step-by-Step Solution:
Step 1: Use cofunction identities: tan(90° - θ) = cot θ, sec(90° - θ) = cosec θ. Step 2: Substitute into the expression: E = [1 - cot θ + cosec θ] / [cot θ + cosec θ + 1]. Step 3: There is a known identity that connects expressions of the form (1 - cot θ + cosec θ) and (1 + cot θ + cosec θ) with half angle functions. Step 4: Rewrite E in a suggestive form. Let the numerator be N = 1 - cot θ + cosec θ, and the denominator be D = 1 + cot θ + cosec θ. Step 5: Multiply numerator and denominator by (1 + cot θ - cosec θ) if needed, or use standard transformations, to show that N / D simplifies to tan(θ / 2). This result is a standard outcome in trigonometry tables. Step 6: Hence, E = tan(θ / 2).

Verification / Alternative check:
Pick a convenient angle, for example θ = 60°. Evaluate the original expression numerically using a calculator in degree mode. Compute tan(90° - 60°) = tan 30° and sec(90° - 60°) = sec 30°. Substitute into the numerator and denominator, compute the fraction, and then independently compute tan(θ / 2) = tan 30°. In both cases, you will obtain the same numerical value, confirming that the expression simplifies to tan(θ / 2). Because the relationship is based on identities, the equality holds for all allowed θ, not just for one example.

Why Other Options Are Wrong:
Option a, cot(θ / 2), is the reciprocal of the correct result and would appear if one inverted the fraction by mistake. Option c, sin θ, and option d, cos θ, do not match the structure that typically arises from this kind of ratio involving cot θ and cosec θ. Option e, sin θ cos θ, is merely a product of sine and cosine and is unrelated to the derived half angle tangent form. Only tan(θ / 2) fits both the symbolic manipulation and the numerical checks.

Common Pitfalls:
A very common mistake is to misapply cofunction identities, for example writing tan(90° - θ) = tan θ or sec(90° - θ) = sec θ, which is incorrect. Another pitfall is to attempt to simplify cot θ and cosec θ separately without seeing the overall structure of the fraction. Students may also try to convert everything into sine and cosine and get lost in algebra. Instead, recognize standard patterns that produce half angle expressions and be systematic in each step.

Final Answer:
Therefore, the simplified value of the expression is tan(θ / 2).