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]
