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Sagot :
Sure, let's establish the trigonometric identity:
[tex]\[ (1 + \tan^2 \theta) \cos^2 \theta = 1 \][/tex]
1. Start with the left-hand side (LHS) of the equation:
[tex]\[ \text{LHS} = (1 + \tan^2 \theta) \cos^2 \theta \][/tex]
2. Recall the Pythagorean identity involving tangent and secant:
[tex]\[ 1 + \tan^2 \theta = \sec^2 \theta \][/tex]
3. Substitute [tex]\(\sec^2 \theta\)[/tex] in place of [tex]\(1 + \tan^2 \theta\)[/tex]:
[tex]\[ \text{LHS} = \sec^2 \theta \cdot \cos^2 \theta \][/tex]
4. Recall the definition of secant. Since [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex], we have:
[tex]\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \][/tex]
5. Substitute [tex]\(\frac{1}{\cos^2 \theta}\)[/tex] for [tex]\(\sec^2 \theta\)[/tex]:
[tex]\[ \text{LHS} = \frac{1}{\cos^2 \theta} \cdot \cos^2 \theta \][/tex]
6. Simplify the expression:
[tex]\[ \frac{1}{\cos^2 \theta} \cdot \cos^2 \theta = 1 \][/tex]
Thus, we have shown that:
[tex]\[ (1 + \tan^2 \theta) \cos^2 \theta = 1 \][/tex]
Therefore, the given identity is established.
[tex]\[ (1 + \tan^2 \theta) \cos^2 \theta = 1 \][/tex]
1. Start with the left-hand side (LHS) of the equation:
[tex]\[ \text{LHS} = (1 + \tan^2 \theta) \cos^2 \theta \][/tex]
2. Recall the Pythagorean identity involving tangent and secant:
[tex]\[ 1 + \tan^2 \theta = \sec^2 \theta \][/tex]
3. Substitute [tex]\(\sec^2 \theta\)[/tex] in place of [tex]\(1 + \tan^2 \theta\)[/tex]:
[tex]\[ \text{LHS} = \sec^2 \theta \cdot \cos^2 \theta \][/tex]
4. Recall the definition of secant. Since [tex]\(\sec \theta = \frac{1}{\cos \theta}\)[/tex], we have:
[tex]\[ \sec^2 \theta = \frac{1}{\cos^2 \theta} \][/tex]
5. Substitute [tex]\(\frac{1}{\cos^2 \theta}\)[/tex] for [tex]\(\sec^2 \theta\)[/tex]:
[tex]\[ \text{LHS} = \frac{1}{\cos^2 \theta} \cdot \cos^2 \theta \][/tex]
6. Simplify the expression:
[tex]\[ \frac{1}{\cos^2 \theta} \cdot \cos^2 \theta = 1 \][/tex]
Thus, we have shown that:
[tex]\[ (1 + \tan^2 \theta) \cos^2 \theta = 1 \][/tex]
Therefore, the given identity is established.
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