Trigonometry Identities: Simplifying Higher Powers

In summary, the given expression can be converted to 1/cos^4 x + sin^2 x/cos^2 x - 2sin^4 x/cos^4 x. Factoring out 1/cos^4 x and simplifying the terms containing sin^2 x, we get cos^2 x + 1 - 2sin^2 x. Applying well known identities, we can further simplify to cos^2 x - sin^2 x + 1. Therefore, the correct answer is d. tan^2 x - 1.
  • #1
NotAMathWhiz
2
0
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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  • #2
Yes, sec^4 x is 1/cos^4 x. Try converting everything in the expression to sines and cosines.
 
  • #3
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)
 
  • #4
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.
 
  • #5
Yes, now simplify that.
 

FAQ: Trigonometry Identities: Simplifying Higher Powers

What are trigonometric identities?

Trigonometric identities are mathematical equations that relate the different trigonometric functions, such as sine, cosine, and tangent, to each other. They are used to simplify and solve trigonometric equations and problems.

What are the most commonly used trigonometric identities?

The most commonly used trigonometric identities include the Pythagorean identities, double angle identities, half angle identities, sum and difference identities, and the reciprocal identities.

How do you prove a trigonometric identity?

To prove a trigonometric identity, you need to manipulate one side of the equation using algebraic and trigonometric properties until it is equivalent to the other side. This can involve using the definitions of the trigonometric functions, common trigonometric identities, and algebraic properties such as factoring and expanding.

What is the difference between an identity and an equation in trigonometry?

An identity is an equation that is true for all values of the variables, while an equation is only true for specific values of the variables. In other words, an identity is always true, while an equation may or may not be true depending on the values of the variables.

Why are trigonometric identities important in science?

Trigonometric identities are important in science because they are used extensively in fields such as physics, engineering, and astronomy to model and solve real-world problems involving angles and triangles. They also help simplify and manipulate complex equations, making them easier to work with and understand.

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