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Sagot :
To solve the expression [tex]\(\left(\sec^2 \theta - 1\right)\left(1 - \csc^2 \theta\right)\)[/tex], let's break it down into detailed steps:
1. Expression for [tex]\(\sec^2 \theta - 1\)[/tex]:
[tex]\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \][/tex]
Therefore,
[tex]\[ \sec^2 \theta - 1 = \frac{1}{\cos^2 \theta} - 1 = \frac{1 - \cos^2 \theta}{\cos^2 \theta} \][/tex]
Using the Pythagorean identity [tex]\( \sin^2 \theta + \cos^2 \theta = 1 \)[/tex], we get:
[tex]\[ 1 - \cos^2 \theta = \sin^2 \theta \][/tex]
Thus,
[tex]\[ \sec^2 \theta - 1 = \frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta \][/tex]
2. Expression for [tex]\(1 - \csc^2 \theta\)[/tex]:
[tex]\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} \][/tex]
Therefore,
[tex]\[ 1 - \csc^2 \theta = 1 - \frac{1}{\sin^2 \theta} = \frac{\sin^2 \theta - 1}{\sin^2 \theta} \][/tex]
Using the Pythagorean identity [tex]\( \sin^2 \theta + \cos^2 \theta = 1 \)[/tex], we get:
[tex]\[ \sin^2 \theta - 1 = -\cos^2 \theta \][/tex]
Thus,
[tex]\[ 1 - \csc^2 \theta = \frac{-\cos^2 \theta}{\sin^2 \theta} = -\cot^2 \theta \][/tex]
3. Combining the results:
We found that:
[tex]\[ \sec^2 \theta - 1 = \tan^2 \theta \][/tex]
[tex]\[ 1 - \csc^2 \theta = -\cot^2 \theta \][/tex]
Therefore,
[tex]\[ \left(\sec^2 \theta - 1\right)\left(1 - \csc^2 \theta\right) = \left(\tan^2 \theta\right) \left(-\cot^2 \theta\right) \][/tex]
4. Simplifying the product:
Recall that [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex] and [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex]. Hence,
[tex]\[ \tan^2 \theta \cdot \cot^2 \theta = \left(\frac{\sin^2 \theta}{\cos^2 \theta} \cdot \frac{\cos^2 \theta}{\sin^2 \theta}\right) = 1 \][/tex]
Therefore,
[tex]\[ \left(\sec^2 \theta - 1\right)\left(1 - \csc^2 \theta\right) = \left(\tan^2 \theta\right) \left(-\cot^2 \theta\right) = \left(\tan^2 \theta\right) \left(-1 \cdot \frac{1}{\tan^2 \theta}\right) = -1 \][/tex]
Thus, the value of [tex]\(\left(\sec^2 \theta - 1\right)\left(1 - \csc^2 \theta\right)\)[/tex] is [tex]\(-1\)[/tex].
1. Expression for [tex]\(\sec^2 \theta - 1\)[/tex]:
[tex]\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \][/tex]
Therefore,
[tex]\[ \sec^2 \theta - 1 = \frac{1}{\cos^2 \theta} - 1 = \frac{1 - \cos^2 \theta}{\cos^2 \theta} \][/tex]
Using the Pythagorean identity [tex]\( \sin^2 \theta + \cos^2 \theta = 1 \)[/tex], we get:
[tex]\[ 1 - \cos^2 \theta = \sin^2 \theta \][/tex]
Thus,
[tex]\[ \sec^2 \theta - 1 = \frac{\sin^2 \theta}{\cos^2 \theta} = \tan^2 \theta \][/tex]
2. Expression for [tex]\(1 - \csc^2 \theta\)[/tex]:
[tex]\[ \csc^2 \theta = \frac{1}{\sin^2 \theta} \][/tex]
Therefore,
[tex]\[ 1 - \csc^2 \theta = 1 - \frac{1}{\sin^2 \theta} = \frac{\sin^2 \theta - 1}{\sin^2 \theta} \][/tex]
Using the Pythagorean identity [tex]\( \sin^2 \theta + \cos^2 \theta = 1 \)[/tex], we get:
[tex]\[ \sin^2 \theta - 1 = -\cos^2 \theta \][/tex]
Thus,
[tex]\[ 1 - \csc^2 \theta = \frac{-\cos^2 \theta}{\sin^2 \theta} = -\cot^2 \theta \][/tex]
3. Combining the results:
We found that:
[tex]\[ \sec^2 \theta - 1 = \tan^2 \theta \][/tex]
[tex]\[ 1 - \csc^2 \theta = -\cot^2 \theta \][/tex]
Therefore,
[tex]\[ \left(\sec^2 \theta - 1\right)\left(1 - \csc^2 \theta\right) = \left(\tan^2 \theta\right) \left(-\cot^2 \theta\right) \][/tex]
4. Simplifying the product:
Recall that [tex]\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)[/tex] and [tex]\(\cot \theta = \frac{\cos \theta}{\sin \theta}\)[/tex]. Hence,
[tex]\[ \tan^2 \theta \cdot \cot^2 \theta = \left(\frac{\sin^2 \theta}{\cos^2 \theta} \cdot \frac{\cos^2 \theta}{\sin^2 \theta}\right) = 1 \][/tex]
Therefore,
[tex]\[ \left(\sec^2 \theta - 1\right)\left(1 - \csc^2 \theta\right) = \left(\tan^2 \theta\right) \left(-\cot^2 \theta\right) = \left(\tan^2 \theta\right) \left(-1 \cdot \frac{1}{\tan^2 \theta}\right) = -1 \][/tex]
Thus, the value of [tex]\(\left(\sec^2 \theta - 1\right)\left(1 - \csc^2 \theta\right)\)[/tex] is [tex]\(-1\)[/tex].
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