sec
4
⁡
𝑥
−
1
tan
2
⁡
𝑥
can be simplified as:

Hint:
sec
2
⁡
𝑥
=
1
+
tan
2
⁡
𝑥

Question

sec
4
⁡
𝑥
−
1
tan
2
⁡
𝑥
can be simplified as:

Hint:  
sec
2
⁡
𝑥
=
1
+
tan
2
⁡
𝑥

BlackTom AI · Accepted Answer

The problem presents a trigonometric simplification task involving secant and tangent expressions. I will evaluate each answer option by applying standard identities and algebraic manipulation, then discuss why each option does or does not align with the given expression.

Option 1: 2 + sec^2 x
- If the original expression simplifies to 2 + sec^2 x, then after simplification the result would have a sec^2 x term plus a constant 2. To verify this, one would expect the algebra to naturally produce a constant term and a sec^2 x term without any other functions (like tan^2 x or cos^2 x) remaining. However, based on common identities (for example, sec^2 x = 1 + tan^2 x), manipulating expressions like sec^4 x − tan^2 x tends to yield forms that involve tan^2 x or sec^2 x in different combinations, and extracting a plain 2 as a constant alongside sec^2 x would require a specific cancellation pattern that is not immediately evident. Without such a cancelation, this option appears dubious.

Option 2: 2 + tan^2 x
- Here, the proposed simplified form includes a tan^2 x term plus a constant 2. Using the identity tan^2 x = sec^2 x − 1, a result of 2 + tan^2 x could be rewritten as 1 + sec^2 x. If the original expression were sec^4 x − tan^2 x, substituting tan^2 x = sec^2 x − 1 would give sec^4 x − (sec^2 x − 1) = sec^4 x − sec^2 x + 1. This does not immediately reduce to 1 + sec^2 x (which would be 2 + tan^2 x). Therefore, unless a hidden simplification exists, this option does not seem to follow directly from standard manipulations.

Option 3: 2 + cos^2 x
- A form with cos^2 x appears unlikely because the original expression uses sec and tan, which are reciprocal/trigonometric complements of cos and sin. Replacing sec or tan with functions involving cos in a way that yields a simple cos^2 x term would require unusual cancellations. Generally, converting sec^4 x or tan^2 x to expressions involving cos^2 x tends to produce 1/cos^4 x or sin^2 x / cos^2 x terms, not a neat cos^2 x addition. Thus this option seems inconsistent with standard simplification routes for sec^4 x − tan^2 x.

Option 4: 1 + cot^2 x
- The identity 1 + cot^2 x = csc^2 x is a valid trig relation, but it's expressed in terms of cot and csc rather than sec and tan. If the target expression were to simplify to 1 + cot^2 x, one would expect an appearance of cot^2 x in the process or a clear path transforming sec and tan into cot and csc. Starting from sec^4 x − tan^2 x, the natural substitutions involve tan^2 x = sec^2 x − 1 (and sec^2 x = 1 + tan^2 x), which keeps the result in terms of sec^2 x and tan^2 x rather than cot^2 x. Therefore, reaching 1 + cot^2 x seems not directly aligned with the usual manipulation of sec^4 x − tan^2 x and appears unlikely as the simplified form.

In summary, each option requires a particular chain of identities to match the original expression. The most straightforward identities to apply here are tan^2 x = sec^2 x − 1 and sec^2 x = 1 + tan^2 x. When these are applied to sec^4 x − tan^2 x, the resulting expression tends to involve sec^4 x, sec^2 x, and 1, rather than cleanly collapsing to a simple 2 + something (as all options propose). Consequently, none of the proposed forms clearly emerges from a direct, standard simplification without additional context or a different starting expression. If the original problem intended a specific target form, providing the exact expression (e.g., sec^4 x − tan^2 x, or sec^4 x − 1 tan^2 x, etc.) would help confirm which option, if any, matches the intended simplification.

sec 4 ⁡ 𝑥 − 1 tan 2 ⁡ 𝑥 can be simplified as: Hint: sec 2 ⁡ 𝑥 = 1 + tan 2 ⁡ 𝑥

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