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.

Simplify the following expression:

[tex]\[ \frac{1 - \cos^4 \theta}{\sin^4 \theta} = 1 + 2 \cot^2 \theta \][/tex]


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.