Trigonometry Identities: Simplifying Higher Powers

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Homework Help Overview

The discussion revolves around simplifying a trigonometric expression involving higher powers of secant and tangent functions. The original poster presents an equation that includes sec^4 x, sec^2 x tan^2 x, and tan^4 x, seeking assistance in simplifying it.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • The original poster expresses confusion about how to begin simplifying the expression, particularly due to their textbook's limited coverage of higher powers. They question whether sec^4 x can be expressed as 1/cos^4 x. Some participants suggest converting the entire expression into sine and cosine terms, while others guide towards factoring and applying known identities.

Discussion Status

Participants are actively engaging with the problem, providing guidance on how to manipulate the expression. There is a collaborative effort to clarify steps and suggest methods for simplification, though no consensus on a final answer has been reached.

Contextual Notes

The original poster notes that their textbook does not cover the topic thoroughly, which may contribute to their uncertainty in handling the problem. The discussion includes attempts to clarify the use of trigonometric identities and the conversion of terms.

NotAMathWhiz
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1. sec^4 x + sec^2 x tan^2 x - 2 tan^4 x = ?
The possible answers are:
a. 4 sec^2 x
b. 3 sec^2 x - 2
c. sec^2 x + 2
d. tan^2 x - 1



Homework Equations



No idea.

The Attempt at a Solution



I'm not sure where to begin here. My book first doesn't cover anything above squared, and when it does, the equation is by itself and immediately shows an easy way to convert to simpler terms. I'm confused however on how to convert this equation.

does sec^4 x = 1/cos^4 x?
 
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Yes, sec^4 x is 1/cos^4 x. Try converting everything in the expression to sines and cosines.
 
okay, so is this correct?

1/cos^4 x + 1/cos^2 x * sin^2 x/cos^2 x - 2(sin^4 x/cos^4 x)
 
Yes. Now factorize out [tex]\frac 1{cos^4x}[/tex] and factorize the terms containing [tex]sin^2x[/tex]. Now, you should be able to apply well known identities to get a numerator that only contains terms involving [tex]cos^2x[/tex]. Now, it should be clear what the answer is.
 
Yes, now simplify that.
 

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