In trigonometric simplification, evaluate the expression: cosec^2 A − cot^2 A + tan^2 A. Express the result in the simplest standard trigonometric form.

Introduction / Context:
This problem tests your knowledge of basic trigonometric identities involving cosec, cot, tan, and sec. The expression is constructed so that standard Pythagoras identities simplify a large part of it. Such questions are common in trigonometry sections to ensure that learners are fluent in converting between different trigonometric functions and know the core relationships among them.

Given Data / Assumptions:
- The expression is cosec^2 A − cot^2 A + tan^2 A.
- The angle A is chosen so that all functions are defined (in particular, sin A and cos A are non zero where needed).
- We are to express the final result in the simplest form, using standard functions like sec^2 A if possible.

Concept / Approach:
There are two key identities to use. First, cosec^2 A − cot^2 A = 1, which is the Pythagoras identity for cosec and cot. Second, 1 + tan^2 A = sec^2 A. The given expression is therefore a direct combination of these two identities. Evaluate the difference in the first two terms to obtain 1, then add tan^2 A and recognise the second identity.

Step-by-Step Solution:
1) Recall the identity for cosecant and cotangent: cosec^2 A − cot^2 A = 1. 2) Apply this directly to the first two terms of the expression, replacing them with 1. 3) The expression now becomes 1 + tan^2 A. 4) Recall the standard identity relating tangent and secant: sec^2 A = 1 + tan^2 A. 5) Substitute this identity to obtain the final simplified form. 6) Therefore, cosec^2 A − cot^2 A + tan^2 A = sec^2 A.

Verification / Alternative check:
You can check this numerically for a simple angle such as A = 45 degrees. Then sin 45° = cos 45° = √2/2, so tan 45° = 1, sec 45° = √2, cosec 45° = √2, and cot 45° = 1. Substituting, we get cosec^2 45° − cot^2 45° + tan^2 45° = (2) − (1) + (1) = 2. On the other hand, sec^2 45° = (√2)^2 = 2. Both sides match, which confirms the identity based simplification.

Why Other Options Are Wrong:
Option A (2 cos 2A) and Option E (1 + cos^2 A) do not follow from the standard Pythagoras identities used here. Option B (1 − sin^2 A) equals cos^2 A, which is different from sec^2 A. Option D (sec A * tan A) is a derivative related identity rather than a Pythagoras type. Only Option C exactly matches the simplified expression 1 + tan^2 A.

Common Pitfalls:
Some students confuse the identity for cosec and cot with that for sec and tan, writing cosec^2 A − cot^2 A as cot^2 A wrongly or mis placing the minus sign. Another common error is to stop at 1 + tan^2 A without recognising that this is identical to sec^2 A. Regular practice with the four core Pythagoras identities in trigonometry helps in remembering and applying them correctly.

Final Answer:
Simplifying using standard identities gives cosec^2 A − cot^2 A + tan^2 A = sec^2 A.