Discover a world of knowledge and get your questions answered at IDNLearn.com. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.
Sagot :
To prove the identity [tex]\(\frac{1 - \cos^4 \theta}{\sin^4 \theta} = 1 + 2 \cot^2 \theta\)[/tex]:
1. Simplify the Left-Hand Side (LHS):
[tex]\[\frac{1 - \cos^4 \theta}{\sin^4 \theta}\][/tex]
We start by expressing [tex]\(1 - \cos^4 \theta\)[/tex] in a factorized form.
Notice that:
[tex]\[1 - \cos^4 \theta = (1 - \cos^2 \theta)(1 + \cos^2 \theta)\][/tex]
Therefore:
[tex]\[\frac{1 - \cos^4 \theta}{\sin^4 \theta} = \frac{(1 - \cos^2 \theta)(1 + \cos^2 \theta)}{\sin^4 \theta}\][/tex]
2. Use the Pythagorean identity:
[tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex].
Therefore, [tex]\(1 - \cos^2 \theta = \sin^2 \theta\)[/tex].
Substituting this in:
[tex]\[\frac{(\sin^2 \theta)(1 + \cos^2 \theta)}{\sin^4 \theta}\][/tex]
Simplify by cancelling [tex]\(\sin^2 \theta\)[/tex] from the numerator and the denominator:
[tex]\[\frac{1 + \cos^2 \theta}{\sin^2 \theta}\][/tex]
3. Rewrite the expression:
[tex]\(\frac{1 + \cos^2 \theta}{\sin^2 \theta}\)[/tex] can be broken down into:
[tex]\[\frac{1}{\sin^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta}\][/tex]
Which simplifies to:
[tex]\(\csc^2 \theta + \cot^2 \theta\)[/tex].
4. Use another Pythagorean identity:
[tex]\(\csc^2 \theta = 1 + \cot^2 \theta\)[/tex].
So:
[tex]\[\csc^2 \theta + \cot^2 \theta = (1 + \cot^2 \theta) + \cot^2 \theta\][/tex]
[tex]\[= 1 + 2 \cot^2 \theta\][/tex]
5. Conclusion:
Therefore:
[tex]\[\frac{1 - \cos^4 \theta}{\sin^4 \theta} = 1 + 2 \cot^2 \theta\][/tex]
Thus, we have shown that the original identity holds true.
1. Simplify the Left-Hand Side (LHS):
[tex]\[\frac{1 - \cos^4 \theta}{\sin^4 \theta}\][/tex]
We start by expressing [tex]\(1 - \cos^4 \theta\)[/tex] in a factorized form.
Notice that:
[tex]\[1 - \cos^4 \theta = (1 - \cos^2 \theta)(1 + \cos^2 \theta)\][/tex]
Therefore:
[tex]\[\frac{1 - \cos^4 \theta}{\sin^4 \theta} = \frac{(1 - \cos^2 \theta)(1 + \cos^2 \theta)}{\sin^4 \theta}\][/tex]
2. Use the Pythagorean identity:
[tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex].
Therefore, [tex]\(1 - \cos^2 \theta = \sin^2 \theta\)[/tex].
Substituting this in:
[tex]\[\frac{(\sin^2 \theta)(1 + \cos^2 \theta)}{\sin^4 \theta}\][/tex]
Simplify by cancelling [tex]\(\sin^2 \theta\)[/tex] from the numerator and the denominator:
[tex]\[\frac{1 + \cos^2 \theta}{\sin^2 \theta}\][/tex]
3. Rewrite the expression:
[tex]\(\frac{1 + \cos^2 \theta}{\sin^2 \theta}\)[/tex] can be broken down into:
[tex]\[\frac{1}{\sin^2 \theta} + \frac{\cos^2 \theta}{\sin^2 \theta}\][/tex]
Which simplifies to:
[tex]\(\csc^2 \theta + \cot^2 \theta\)[/tex].
4. Use another Pythagorean identity:
[tex]\(\csc^2 \theta = 1 + \cot^2 \theta\)[/tex].
So:
[tex]\[\csc^2 \theta + \cot^2 \theta = (1 + \cot^2 \theta) + \cot^2 \theta\][/tex]
[tex]\[= 1 + 2 \cot^2 \theta\][/tex]
5. Conclusion:
Therefore:
[tex]\[\frac{1 - \cos^4 \theta}{\sin^4 \theta} = 1 + 2 \cot^2 \theta\][/tex]
Thus, we have shown that the original identity holds true.
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Find clear answers at IDNLearn.com. Thanks for stopping by, and come back for more reliable solutions.