At Westonci.ca, we provide clear, reliable answers to all your questions. Join our vibrant community and get the solutions you need. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To solve the problem of finding which expression is equal to [tex]\( 1 - \cos^4 \theta \)[/tex], let us start by breaking down the expression and simplifying it step-by-step.
1. Expression Parsing:
We start with the function [tex]\( 1 - \cos^4 \theta \)[/tex].
2. Use the Pythagorean Identity:
Recall the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
From this, we can express [tex]\(\cos^2 \theta\)[/tex] as:
[tex]\[ \cos^2 \theta = 1 - \sin^2 \theta \][/tex]
3. Square the Cosine Expression:
Next, let’s square [tex]\( \cos^2 \theta \)[/tex] to get [tex]\(\cos^4 \theta \)[/tex]:
[tex]\[ \cos^4 \theta = (1 - \sin^2 \theta)^2 \][/tex]
4. Expand the Squared Expression:
Now, let's expand [tex]\( (1 - \sin^2 \theta)^2 \)[/tex]:
[tex]\[ (1 - \sin^2 \theta)^2 = 1 - 2\sin^2 \theta + \sin^4 \theta \][/tex]
5. Substitute Back into the Original Expression:
Substitute the expanded form back into the original expression [tex]\( 1 - \cos^4 \theta \)[/tex]:
[tex]\[ 1 - \cos^4 \theta = 1 - (1 - 2\sin^2 \theta + \sin^4 \theta) \][/tex]
6. Simplify the Expression:
Simplify the expression by distributing the negative sign and combining like terms:
[tex]\[ 1 - (1 - 2\sin^2 \theta + \sin^4 \theta) = 1 - 1 + 2\sin^2 \theta - \sin^4 \theta \][/tex]
[tex]\[ = 2\sin^2 \theta - \sin^4 \theta \][/tex]
Therefore, the expression that is equal to [tex]\( 1 - \cos^4 \theta \)[/tex] is:
[tex]\[ 2\sin^2 \theta - \sin^4 \theta \][/tex]
So, the correct option is:
[tex]\[ \boxed{2 \sin^2 \theta - \sin^4 \theta} \][/tex]
1. Expression Parsing:
We start with the function [tex]\( 1 - \cos^4 \theta \)[/tex].
2. Use the Pythagorean Identity:
Recall the Pythagorean identity:
[tex]\[ \sin^2 \theta + \cos^2 \theta = 1 \][/tex]
From this, we can express [tex]\(\cos^2 \theta\)[/tex] as:
[tex]\[ \cos^2 \theta = 1 - \sin^2 \theta \][/tex]
3. Square the Cosine Expression:
Next, let’s square [tex]\( \cos^2 \theta \)[/tex] to get [tex]\(\cos^4 \theta \)[/tex]:
[tex]\[ \cos^4 \theta = (1 - \sin^2 \theta)^2 \][/tex]
4. Expand the Squared Expression:
Now, let's expand [tex]\( (1 - \sin^2 \theta)^2 \)[/tex]:
[tex]\[ (1 - \sin^2 \theta)^2 = 1 - 2\sin^2 \theta + \sin^4 \theta \][/tex]
5. Substitute Back into the Original Expression:
Substitute the expanded form back into the original expression [tex]\( 1 - \cos^4 \theta \)[/tex]:
[tex]\[ 1 - \cos^4 \theta = 1 - (1 - 2\sin^2 \theta + \sin^4 \theta) \][/tex]
6. Simplify the Expression:
Simplify the expression by distributing the negative sign and combining like terms:
[tex]\[ 1 - (1 - 2\sin^2 \theta + \sin^4 \theta) = 1 - 1 + 2\sin^2 \theta - \sin^4 \theta \][/tex]
[tex]\[ = 2\sin^2 \theta - \sin^4 \theta \][/tex]
Therefore, the expression that is equal to [tex]\( 1 - \cos^4 \theta \)[/tex] is:
[tex]\[ 2\sin^2 \theta - \sin^4 \theta \][/tex]
So, the correct option is:
[tex]\[ \boxed{2 \sin^2 \theta - \sin^4 \theta} \][/tex]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.