Simplify the trigonometric expression (1 + sec 20° + cot 70°)(1 − cosec 20° + tan 70°) and find its exact numerical value.

Introduction / Context:
This question examines simplification of trigonometric expressions using complementary angles and basic identities. The expression mixes sec, cosec, tan and cot at angles 20° and 70°, and you must reduce it to a single numeric value. Such questions test recognition of relationships like tan(90° − θ) = cot θ and sin(90° − θ) = cos θ.

Given Data / Assumptions:

• Expression: (1 + sec 20° + cot 70°)(1 − cosec 20° + tan 70°).
• Angles are in degrees.
• Basic identities: tan(90° − θ) = cot θ and cot(90° − θ) = tan θ.
• Definitions: tan θ = sin θ / cos θ, cot θ = cos θ / sin θ, sec θ = 1 / cos θ, cosec θ = 1 / sin θ.

Concept / Approach:
We first express cot 70° and tan 70° in terms of 20° using complementary angle identities. Then we simplify the product by grouping like terms and using the relations between sec, cosec, sine and cosine. The goal is to reduce everything to simple combinations of sin 20° and cos 20°, which ultimately cancel to yield a small integer.

Step-by-Step Solution:
Use complementary angles: cot 70° = tan(20°) and tan 70° = cot(20°). Rewrite the expression as: (1 + sec 20° + tan 20°)(1 − cosec 20° + cot 20°). Let s = sin 20° and c = cos 20° for brevity. Then sec 20° = 1/c, cosec 20° = 1/s, tan 20° = s/c, and cot 20° = c/s. So the product becomes: (1 + 1/c + s/c)(1 − 1/s + c/s). Factor common denominators in each bracket: First bracket: 1 + (1 + s)/c. Second bracket: 1 + (c − 1)/s. Now expand carefully and rewrite all terms over common denominators so that cancellations using s^2 + c^2 = 1 occur. After systematic simplification, many terms cancel, and the final result reduces to 2.
Verification / Alternative check:
You can pick a numerical approximation of 20° (in radians) and evaluate both brackets on a calculator with adequate precision. The product will be extremely close to 2, confirming the algebraic simplification.

Why Other Options Are Wrong:
A result of 0 or 1 would require nearly complete cancellation between the two brackets, which does not happen once you correctly use the complementary identities. Values like 3 or 4 appear if terms are added incorrectly or if complementary angle formulas are misapplied.

Common Pitfalls:
The main mistakes come from misusing tan 70° and cot 70°, or mixing up s and c when substituting. Another frequent issue is incomplete simplification: stopping at a complicated fraction instead of fully cancelling common factors. Working systematically with s and c, and keeping careful track of each term, avoids these errors.

Final Answer:
The simplified value of the expression is 2.