Given:

\(\displaystyle{{\cot}^{{2}}\theta}-{{\cos}^{{2}}\theta}={{\cot}^{{2}}\theta}{{\cos}^{\theta}}\)

Manipulating right side

\(\displaystyle{{\cot}^{{2}}\theta}{{\cos}^{{2}}\theta}\) Rewrite using trigonometry identities: \(\displaystyle{{\cos}^{{2}}\theta}={1}-{{\sin}^{{2}}\theta}\)

\(\displaystyle={\left({1}-{{\sin}^{{2}}\theta}\right)}{{\cot}^{{2}}\theta}\)

Expand \(\displaystyle{\left({1}-{{\sin}^{{2}}\theta}\right)}{{\cot}^{{2}}\theta}:{{\cot}^{{2}}\theta}-{{\cot}^{{2}}\theta}{{\sin}^{{2}}\theta}\)

\(\displaystyle={{\cot}^{{2}}\theta}-{{\cot}^{{2}}\theta}{{\sin}^{{2}}\theta}\)

Simplify using: \(\displaystyle{{\cot}^{{2}}\theta}={\frac{{{{\cos}^{{2}}\theta}}}{{{{\sin}^{{2}}\theta}}}}\)

\(\displaystyle={{\cot}^{{2}}\theta}-{\frac{{{{\cos}^{{2}}\theta}}}{{{\sin}^{\theta}}}}\times{{\sin}^{{2}}\theta}\)

\(\displaystyle={{\cot}^{{2}}\theta}-{{\cos}^{\theta}}\)

We showed that the two sides could take the same form.

Hence proved

\(\displaystyle{{\cot}^{{2}}\theta}-{{\cos}^{{2}}\theta}={{\cot}^{{2}}\theta}{{\cos}^{\theta}}\)

Manipulating right side

\(\displaystyle{{\cot}^{{2}}\theta}{{\cos}^{{2}}\theta}\) Rewrite using trigonometry identities: \(\displaystyle{{\cos}^{{2}}\theta}={1}-{{\sin}^{{2}}\theta}\)

\(\displaystyle={\left({1}-{{\sin}^{{2}}\theta}\right)}{{\cot}^{{2}}\theta}\)

Expand \(\displaystyle{\left({1}-{{\sin}^{{2}}\theta}\right)}{{\cot}^{{2}}\theta}:{{\cot}^{{2}}\theta}-{{\cot}^{{2}}\theta}{{\sin}^{{2}}\theta}\)

\(\displaystyle={{\cot}^{{2}}\theta}-{{\cot}^{{2}}\theta}{{\sin}^{{2}}\theta}\)

Simplify using: \(\displaystyle{{\cot}^{{2}}\theta}={\frac{{{{\cos}^{{2}}\theta}}}{{{{\sin}^{{2}}\theta}}}}\)

\(\displaystyle={{\cot}^{{2}}\theta}-{\frac{{{{\cos}^{{2}}\theta}}}{{{\sin}^{\theta}}}}\times{{\sin}^{{2}}\theta}\)

\(\displaystyle={{\cot}^{{2}}\theta}-{{\cos}^{\theta}}\)

We showed that the two sides could take the same form.

Hence proved