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Sagot :
Sure, let’s go through a detailed, step-by-step solution to simplify the given expression:
[tex]\[ \frac{(\cos x)^2}{(\cot x)^2} \][/tex]
1. Recall the definition of [tex]\(\cot x\)[/tex]:
[tex]\[ \cot x = \frac{\cos x}{\sin x} \][/tex]
2. Square both sides of the identity for [tex]\(\cot x\)[/tex]:
[tex]\[ (\cot x)^2 = \left(\frac{\cos x}{\sin x}\right)^2 = \frac{(\cos x)^2}{(\sin x)^2} \][/tex]
3. Substitute the squared expression for [tex]\((\cot x)^2\)[/tex] into the original expression:
[tex]\[ \frac{(\cos x)^2}{(\cot x)^2} = \frac{(\cos x)^2}{\frac{(\cos x)^2}{(\sin x)^2}} \][/tex]
4. Simplify the fraction:
[tex]\[ \frac{(\cos x)^2}{\frac{(\cos x)^2}{(\sin x)^2}} = (\cos x)^2 \cdot \frac{(\sin x)^2}{(\cos x)^2} \][/tex]
5. Cancel out the common factor of [tex]\((\cos x)^2\)[/tex] in the numerator and the denominator:
[tex]\[ = \frac{(\cos x)^2 \cdot (\sin x)^2}{(\cos x)^2} = (\sin x)^2 \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ (\sin x)^2 \][/tex]
[tex]\[ \frac{(\cos x)^2}{(\cot x)^2} \][/tex]
1. Recall the definition of [tex]\(\cot x\)[/tex]:
[tex]\[ \cot x = \frac{\cos x}{\sin x} \][/tex]
2. Square both sides of the identity for [tex]\(\cot x\)[/tex]:
[tex]\[ (\cot x)^2 = \left(\frac{\cos x}{\sin x}\right)^2 = \frac{(\cos x)^2}{(\sin x)^2} \][/tex]
3. Substitute the squared expression for [tex]\((\cot x)^2\)[/tex] into the original expression:
[tex]\[ \frac{(\cos x)^2}{(\cot x)^2} = \frac{(\cos x)^2}{\frac{(\cos x)^2}{(\sin x)^2}} \][/tex]
4. Simplify the fraction:
[tex]\[ \frac{(\cos x)^2}{\frac{(\cos x)^2}{(\sin x)^2}} = (\cos x)^2 \cdot \frac{(\sin x)^2}{(\cos x)^2} \][/tex]
5. Cancel out the common factor of [tex]\((\cos x)^2\)[/tex] in the numerator and the denominator:
[tex]\[ = \frac{(\cos x)^2 \cdot (\sin x)^2}{(\cos x)^2} = (\sin x)^2 \][/tex]
Thus, the simplified form of the given expression is:
[tex]\[ (\sin x)^2 \][/tex]
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