Evaluate: \lim_{x arrow (pi)/(2)} ((3cos(x)^4)/(cot(x)^2-cos(x)^2))

Expression: $\lim_{x \rightarrow \frac{ π }{ 2 }} \left(\frac{ 3{\cos\left({x}\right)}^{4} }{ {\cot\left({x}\right)}^{2}-{\cos\left({x}\right)}^{2} }\right)$

Since evaluating limits of the numerator and denominator would result in an indeterminate form, use the L'Hopital's rule

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(\frac{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( 3{\cos\left({x}\right)}^{4} \right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( {\cot\left({x}\right)}^{2}-{\cos\left({x}\right)}^{2} \right) }\right)$

Find the derivative

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(\frac{ -12{\cos\left({x}\right)}^{3}\sin\left({x}\right) }{ \frac{ \mathrm{d} }{ \mathrm{d}x} \left( {\cot\left({x}\right)}^{2}-{\cos\left({x}\right)}^{2} \right) }\right)$

Find the derivative

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(\frac{ -12{\cos\left({x}\right)}^{3}\sin\left({x}\right) }{ -\frac{ 2\cos\left({x}\right) }{ {\sin\left({x}\right)}^{3} }+\sin\left({2x}\right) }\right)$

Write all numerators above the common denominator

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(\frac{ -12{\cos\left({x}\right)}^{3}\sin\left({x}\right) }{ \frac{ -2\cos\left({x}\right)+{\sin\left({x}\right)}^{3}\sin\left({2x}\right) }{ {\sin\left({x}\right)}^{3} } }\right)$

Use $\frac{ -a }{ b }=\frac{ a }{ -b }=-\frac{ a }{ b }$ to rewrite the fraction

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ 12{\cos\left({x}\right)}^{3}\sin\left({x}\right) }{ \frac{ -2\cos\left({x}\right)+{\sin\left({x}\right)}^{3}\sin\left({2x}\right) }{ {\sin\left({x}\right)}^{3} } }\right)$

Simplify the complex fraction

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ 12{\cos\left({x}\right)}^{3}{\sin\left({x}\right)}^{4} }{ -2\cos\left({x}\right)+{\sin\left({x}\right)}^{3}\sin\left({2x}\right) }\right)$

Use $\sin\left({2t}\right)=2\sin\left({t}\right)\cos\left({t}\right)$ to expand the expression

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ 12{\cos\left({x}\right)}^{3}{\sin\left({x}\right)}^{4} }{ -2\cos\left({x}\right)+{\sin\left({x}\right)}^{3} \times 2\sin\left({x}\right)\cos\left({x}\right) }\right)$

Factor out $-2\cos\left({x}\right)$ from the expression

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ 12{\cos\left({x}\right)}^{3}{\sin\left({x}\right)}^{4} }{ -2\cos\left({x}\right) \times \left( 1-{\sin\left({x}\right)}^{3}\sin\left({x}\right) \right) }\right)$

Cancel out the common factor $-2$

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\cos\left({x}\right)}^{3}{\sin\left({x}\right)}^{4} }{ \cos\left({x}\right) \times \left( 1-{\sin\left({x}\right)}^{3}\sin\left({x}\right) \right) }\right)$

Cancel out the common factor $\cos\left({x}\right)$

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\cos\left({x}\right)}^{2}{\sin\left({x}\right)}^{4} }{ 1-{\sin\left({x}\right)}^{3}\sin\left({x}\right) }\right)$

Calculate the product

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\cos\left({x}\right)}^{2}{\sin\left({x}\right)}^{4} }{ 1-{\sin\left({x}\right)}^{4} }\right)$

Use ${a}^{2}-{b}^{2}=\left( a-b \right)\left( a+b \right)$ to factor the expression

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\cos\left({x}\right)}^{2}{\sin\left({x}\right)}^{4} }{ \left( 1-{\sin\left({x}\right)}^{2} \right) \times \left( 1+{\sin\left({x}\right)}^{2} \right) }\right)$

Use $1-{\sin\left({t}\right)}^{2}={\cos\left({t}\right)}^{2}$ to simplify the expression

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\cos\left({x}\right)}^{2}{\sin\left({x}\right)}^{4} }{ {\cos\left({x}\right)}^{2} \times \left( 1+{\sin\left({x}\right)}^{2} \right) }\right)$

Cancel out the common factor ${\cos\left({x}\right)}^{2}$

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(-\frac{ -6{\sin\left({x}\right)}^{4} }{ 1+{\sin\left({x}\right)}^{2} }\right)$

Determine the sign of the fraction

$\lim_{x \rightarrow \frac{ π }{ 2 }} \left(\frac{ 6{\sin\left({x}\right)}^{4} }{ 1+{\sin\left({x}\right)}^{2} }\right)$

Evaluate the limit

$\frac{ 6{\sin\left({\frac{ π }{ 2 }}\right)}^{4} }{ 1+{\sin\left({\frac{ π }{ 2 }}\right)}^{2} }$

Simplify the expression

$3$

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