- #1
General_Sax
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Homework Statement
Must be proven algebraically, duh!
Homework Equations
trig identities
The Attempt at a Solution
I'm at a loss as what to do next. Any help would be appreciated.
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General_Sax said:Let me restart from here:
[tex] \frac{2 - [cos^2x ( 1 + 2(1 - sin^2x)]}{2sin^4x} = \frac{1 - cos^2x}{sin^2x}[/tex]
[tex] \frac{2 - [cos^2x ( 1 + 2 - 2sin^2x)]}{2sin^4x} = \frac{1 - cos^2x}{sin^2x}[/tex]
[tex] \frac{2 - [3cos^2x - 2sin^2xcos^2x]}{2sin^4x} = \frac{1 - cos^2x}{sin^2x}[/tex]
[tex]\frac{2 - 3cos^2x + 2cos^2x}{2sin^2x} = \frac{1 - cos^2x}{sin^2x}[/tex]
In the previous step I subtracted the exponents for sin in the denominator from the numerator. Is this a legal maneuver?
[tex]\frac{2 - 2cos^2x}{2sin^2x} = \frac{1 - cos^2x}{sin^2x}[/tex]
[tex]\frac{2(1-cos^2x)}{2sin^2x} = \frac{1 - cos^2x}{sin^2x}[/tex]
[tex]\frac{1-cos^2x}{sin^2x} = \frac{1 - cos^2x}{sin^2x}[/tex]
Is this correct? Thanks again for the help guys.
An identity in algebra is a mathematical statement that is always true, no matter what values are substituted for the variables. It is a way to show that two expressions are equivalent.
To prove an identity algebraically, you need to manipulate one side of the equation using the properties of algebra until it is equivalent to the other side. This can include using the commutative, associative, and distributive properties, as well as combining like terms and factoring.
Common mistakes when proving an identity algebraically include not following the order of operations, making incorrect algebraic manipulations, and not checking both sides of the equation to ensure they are equivalent.
Yes, you can use numerical examples to help understand and demonstrate an identity, but it is not a valid proof. A proof must use algebraic manipulations to show that the identity holds for all values of the variables.
Proving identities is important in mathematics because it helps to strengthen our understanding of algebraic principles and allows us to solve more complex equations. It also helps to verify the validity of mathematical equations and ensures that they are always true.